You may want to refer to this answer to the question "Is the Quran complete? If so, why do we need Hadith?" on Islam Stack Exchange.
The Qur'an has a verse that explicitly says it is the Book about which there is no doubt:
ذَٰلِكَ الْكِتَابُ لَا رَيْبَ ۛ فِيهِ ۛ هُدًى لِّلْمُتَّقِينَ
This is the Book about which there is no doubt, a guidance for those conscious of Allah.
— Surat Al-Baqarah 2:2
However, it is referring to the entire book, not just this verse. It is in this book that Allah mandated that we obey the teachings of the Prophet ﷺ and considered this obedience an essential part of our faith without which we become disbelievers:
فَلَا وَرَبِّكَ لَا يُؤْمِنُونَ حَتَّىٰ يُحَكِّمُوكَ فِيمَا شَجَرَ بَيْنَهُمْ ثُمَّ لَا يَجِدُوا فِي أَنفُسِهِمْ حَرَجًا مِّمَّا قَضَيْتَ وَيُسَلِّمُوا تَسْلِيمًا
But no, by your Lord, they will not [truly] believe until they make you, [O Muhammad], judge concerning that over which they dispute among themselves and then find within themselves no discomfort from what you have judged and submit in [full, willing] submission.
— Surat An-Nisa 4:65
The verse above is not the only verse in the Qur'an that referred to the Prophet ﷺ as our judge in this world and as our mediator. Another verse also stipulates this as a condition of belief in Allah and Judgment Day:
يَا أَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ وَأُولِي الْأَمْرِ مِنكُمْ ۖ فَإِن تَنَازَعْتُمْ فِي شَيْءٍ فَرُدُّوهُ إِلَى اللَّهِ وَالرَّسُولِ إِن كُنتُمْ تُؤْمِنُونَ بِاللَّهِ وَالْيَوْمِ الْآخِرِ ۚ ذَٰلِكَ خَيْرٌ وَأَحْسَنُ تَأْوِيلًا
O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result.
— Surat An-Nisa 4:59
The Prophet ﷺ is no longer with us, and we definitely cannot refer to him regarding our disputes, but we can refer to how he handled similar matters and use this as our guidance — this is called his Sunnah. So belief in the Qur'an automatically leads to belief in the Sunnah; not only so, it leads to disbelief in the Sunnah to equate to disbelief in the Qur'an, and Allah and Judgment Day as the Qur'an itself says.
Moreover, the Sunnah of the Prophet ﷺ is a revelation from Allah ﷻ as is the Qur'an:
وَلَوْلَا فَضْلُ اللَّهِ عَلَيْكَ وَرَحْمَتُهُ لَهَمَّت طَّائِفَةٌ مِّنْهُمْ أَن يُضِلُّوكَ وَمَا يُضِلُّونَ إِلَّا أَنفُسَهُمْ ۖ وَمَا يَضُرُّونَكَ مِن شَيْءٍ ۚ وَأَنزَلَ اللَّهُ عَلَيْكَ الْكِتَابَ وَالْحِكْمَةَ وَعَلَّمَكَ مَا لَمْ تَكُن تَعْلَمُ وَكَانَ فَضْلُ اللَّهِ عَلَيْكَ عَظِيمًا
And if it was not for the favor of Allah upon you, [O Muhammad], and His mercy, a group of them would have determined to mislead you. But they do not mislead except themselves, and they will not harm you at all. And Allah has revealed to you the Book and wisdom and has taught you that which you did not know. And ever has the favor of Allah upon you been great.
— Surat An-Nisa 4:113
We can all agree that the Book is the Qur'an. What is the wisdom that Allah has revealed? It is the Sunnah. Even if one wants to deny so, one would have to acknowledge that said wisdom is what is contained in the Book (Qur'an); be that as it may, it would have demonstrated to us through what the Prophet ﷺ said or did during his life, i.e., his Sunnah. Furthermore, this is what scholars have deduced about the word "wisdom" in the Qur'an:
قال الشَّافِعِي رحمه الله تعالى: قال اللَّه تبارك وتعالى: (وَأَنْزَلَ اللَّهُ عَلَيْكَ الْكِتَابَ وَالْحِكْمَةَ وَعَلَّمَكَ مَا لَمْ تَكُنْ تَعْلَمُ) فيذهب إلى أن الكتاب هو ما يتلى عن اللَّه تعالى والحكمة هي ما جاءت به الرسالة عن اللَّه، مما بينْت سُنَّة لرسول الله ﷺ
NOTE. My own translation, so treat with care.
Ash-Shāfi'i — may Allah have mercy on him — said: "Allah, may He be Blessed and Exalted, said: 'and Allah has revealed to you the Book and wisdom and has taught you that which you did not know.' It is deduced that the Book is what is recited from [the words of] Allah, and the wisdom is what the message came with from Allah, which was clarified through the Sunnah of his Prophet ﷺ."
— Tafsīr al-Imām Ash-Shāfi'i, Vol. 8, pp. 668 (Arabic only)
There are numerous uses of the letter "waw" (Arabic: و) in the Arabic language, one of which is to mean "and", and as linguistics say it indicates a difference (Arabic: المغايرة) meaning that what precedes it is not the same as what follows it. This means that the Book and the wisdom are not the same things, and this was mentioned more than once in the Book:
هُوَ الَّذِي بَعَثَ فِي الْأُمِّيِّينَ رَسُولًا مِّنْهُمْ يَتْلُو عَلَيْهِمْ آيَاتِهِ وَيُزَكِّيهِمْ وَيُعَلِّمُهُمُ الْكِتَابَ وَالْحِكْمَةَ وَإِن كَانُوا مِن قَبْلُ لَفِي ضَلَالٍ مُّبِينٍ
It is He who has sent among the unlettered a Messenger from themselves reciting to them His verses and purifying them and teaching them the Book and wisdom — although they were before in clear error —
— Surat Al-Jumu'ah 62:2
كَمَا أَرْسَلْنَا فِيكُمْ رَسُولًا مِّنكُمْ يَتْلُو عَلَيْكُمْ آيَاتِنَا وَيُزَكِّيكُمْ وَيُعَلِّمُكُمُ الْكِتَابَ وَالْحِكْمَةَ وَيُعَلِّمُكُم مَّا لَمْ تَكُونُوا تَعْلَمُونَ
Just as We have sent among you a messenger from yourselves reciting to you Our verses and purifying you and teaching you the Book and wisdom and teaching you that which you did not know.
— Surat Al-Baqarah 2:151
وَاذْكُرُوا نِعْمَتَ اللَّهِ عَلَيْكُمْ وَمَا أَنزَلَ عَلَيْكُم مِّنَ الْكِتَابِ وَالْحِكْمَةِ يَعِظُكُم بِهِ ۚ وَاتَّقُوا اللَّهَ وَاعْلَمُوا أَنَّ اللَّهَ بِكُلِّ شَيْءٍ عَلِيمٌ
And remember the favor of Allah upon you and what has been revealed to you of the Book and wisdom by which He instructs you. And fear Allah and know that Allah is Knowing of all things.
— Surat Al-Baqarah 2:231
So, if one is to follow the Qur'an alone, it would automatically lead to following the Sunnah, too, unless one elects to be among the ignorant, the ones who are in clear error, or the ones who have gone astray.
There is nothing related to what the Prophet ﷺ said or did in relation to the religion except through a revelation from Allah ﷻ:
وَمَا يَنطِقُ عَنِ الْهَوَىٰ إِنْ هُوَ إِلَّا وَحْيٌ يُوحَىٰ
Nor does he speak from [his own] inclination. It is not but a revelation revealed,
— Surat An-Najm 53:3-4
If one were to deny what the Prophet ﷺ said or did, one would be denying the Book, too. The only way out of this, logically, is to reject what the Prophet ﷺ which means rejecting the Book, too; or to accept the Book, and accordingly obey the Prophet ﷺ as this is what the Book says:
يَا أَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ وَلَا تُبْطِلُوا أَعْمَالَكُمْ
O you who have believed, obey Allah and obey the Messenger and do not invalidate your deeds.
— Surat Muhammad 47:33
Not obeying the Prophet ﷺ renders our deeds invalid as the verse above says. We also run the risk of going astray if we do not follow his Sunnah, according to the Book, or to become disbelievers, or to disobey Allah and turn away from Him, or miss His mercy, etc.:
قُلْ أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ ۖ فَإِن تَوَلَّوْا فَإِنَّمَا عَلَيْهِ مَا حُمِّلَ وَعَلَيْكُم مَّا حُمِّلْتُمْ ۖ وَإِن تُطِيعُوهُ تَهْتَدُوا ۚ وَمَا عَلَى الرَّسُولِ إِلَّا الْبَلَاغُ الْمُبِينُ
Say, "Obey Allah and obey the Messenger; but if you turn away - then upon him is only that [duty] with which he has been charged, and upon you is that with which you have been charged. And if you obey him, you will be [rightly] guided. And there is not upon the Messenger except the [responsibility for] clear notification."
— Surat An-Nur 24:54
قُلْ أَطِيعُوا اللَّهَ وَالرَّسُولَ ۖ فَإِن تَوَلَّوْا فَإِنَّ اللَّهَ لَا يُحِبُّ الْكَافِرِينَ
Say, "Obey Allah and the Messenger." But if they turn away - then indeed, Allah does not like the disbelievers.
— Surat Ali 'Imran 3:32
يَا أَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَرَسُولَهُ وَلَا تَوَلَّوْا عَنْهُ وَأَنتُمْ تَسْمَعُونَ
O you who have believed, obey Allah and His Messenger and do not turn from him while you hear [his order].
— Surat Al-Anfal 8:20
وَأَقِيمُوا الصَّلَاةَ وَآتُوا الزَّكَاةَ وَأَطِيعُوا الرَّسُولَ لَعَلَّكُمْ تُرْحَمُونَ
And establish prayer and give zakah and obey the Messenger - that you may receive mercy.
— Surat An-Nur 24:56
مَّن يُطِعِ الرَّسُولَ فَقَدْ أَطَاعَ اللَّهَ ۖ وَمَن تَوَلَّىٰ فَمَا أَرْسَلْنَاكَ عَلَيْهِمْ حَفِيظًا
He who obeys the Messenger has obeyed Allah; but those who turn away - We have not sent you over them as a guardian.
— Surat An-Nisa 4:80
Not only so, but Allah has in numerous verses in the Qur'an tell us that disobeying Him will only mean we may end up in Hellfire. Likewise, Allah ﷻ said the same thing about disobeying the Prophet ﷺ without even explicitly referring to disobedience of Allah. This is because disobedience of the Prophet ﷺ (his Sunnah) is synonymous to disobedience of Allah:
يَوْمَئِذٍ يَوَدُّ الَّذِينَ كَفَرُوا وَعَصَوُا الرَّسُولَ لَوْ تُسَوَّىٰ بِهِمُ الْأَرْضُ وَلَا يَكْتُمُونَ اللَّهَ حَدِيثًا
That Day, those who disbelieved and disobeyed the Messenger will wish they could be covered by the earth. And they will not conceal from Allah a [single] statement.
— Surat An-Nisa 4:42
The Prophet ﷺ even prophecised about this in his hadiths, ironically enough to those who claim that the Sunnah is not to be followed:
عن عبيد الله بن أبي رافع عن أبيه أن رسول الله صلى الله عليه وسلم قال:
لاَ أُلْفِيَنَّ أَحَدَكُمْ مُتَّكِئًا عَلَى أَرِيكَتِهِ يَأْتِيهِ الأَمْرُ مِمَّا أَمَرْتُ بِهِ أَوْ نَهَيْتُ عَنْهُ فَيَقُولُ لاَ أَدْرِي مَا وَجَدْنَا فِي كِتَابِ اللَّهِ اتَّبَعْنَاهُ
It was narrated from 'Ubaidullah bin Abu Rafi from his father, that: The Messenger of Allah (ﷺ) said: "I do not want to find anyone of you reclining on his pillow, and when bad news comes to him of something that I have commanded or forbidden, he says, 'I do not know, whatever we find in the Book of Allah, we will follow."
— Sunan Ibn Majah, Vol. 1, Book 1, Hadith 13
The selection of the words by the Prophet ﷺ are just amazing; "reclining on his pillow" is very indicative of what is indeed taking place today. But, there are also additional rulings that are in the hadith that complement the Qur'an, for instance:
عن المقدام بن معديكرب عن رسول الله صلى الله عليه وسلم أنه قال: أَلاَ إِنِّي أُوتِيتُ الْكِتَابَ وَمِثْلَهُ مَعَهُ أَلاَ يُوشِكُ رَجُلٌ شَبْعَانُ عَلَى أَرِيكَتِهِ يَقُولُ عَلَيْكُمْ بِهَذَا الْقُرْآنِ فَمَا وَجَدْتُمْ فِيهِ مِنْ حَلاَلٍ فَأَحِلُّوهُ وَمَا وَجَدْتُمْ فِيهِ مِنْ حَرَامٍ فَحَرِّمُوهُ أَلاَ لاَ يَحِلُّ لَكُمْ لَحْمُ الْحِمَارِ الأَهْلِيِّ وَلاَ كُلُّ ذِي نَابٍ مِنَ السَّبُعِ وَلاَ لُقَطَةُ مُعَاهِدٍ إِلاَّ أَنْ يَسْتَغْنِيَ عَنْهَا صَاحِبُهَا وَمَنْ نَزَلَ بِقَوْمٍ فَعَلَيْهِمْ أَنْ يَقْرُوهُ فَإِنْ لَمْ يَقْرُوهُ فَلَهُ أَنْ يُعْقِبَهُمْ بِمِثْلِ قِرَاهُ
Narrated Al-Miqdam ibn Ma'dikarib: The Prophet (ﷺ) said: Beware! I have been given the Qur'an and something like it, yet the time is coming when a man replete on his couch will say: Keep to the Qur'an; what you find in it to be permissible treat as permissible, and what you find in it to be prohibited treat as prohibited. Beware! The domestic ass, beasts of prey with fangs, a find belonging to confederate, unless its owner does not want it, are not permissible to you If anyone comes to some people, they must entertain him, but if they do not, he has a right to mulct them to an amount equivalent to his entertainment.
— Sunan Abi Dawud, Book 42, Hadith 9
Some may argue that this means the Qur'an is not comprehensive when it comes to what to follow and not to follow. This is incorrect as the Qur'an very clearly instructs us to obey the Prophet ﷺ:
يَا أَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ وَلَا تُبْطِلُوا أَعْمَالَكُمْ
O you who have believed, obey Allah and obey the Messenger and do not invalidate your deeds.
— Surat Muhammad 47:33
So anytime one obeys the Prophet ﷺ, one is abiding by what is in the Qur'an. I am not going to get into the religious rites that cannot be fulfilled without hadith (e.g., how many rak'ahs the prayer of maghrib is, or how should one pray in the first place, etc.).