The Qur'an contains many verses that are either addressed to or intended for various categories of people.
Some examples:
Mankind in general
يَا أَيُّهَا النَّاسُ اتَّقُوا رَبَّكُمْ إِنَّ زَلْزَلَةَ السَّاعَةِ شَيْءٌ عَظِيمٌ
O mankind! Fear your Lord and be dutiful to Him! Verily, the earthquake of the Hour (of Judgement) is a terrible thing.
Al-Hajj:1Jews
قُلْ يَا أَيُّهَا الَّذِينَ هَادُوا إِن زَعَمْتُمْ أَنَّكُمْ أَوْلِيَاءُ لِلَّـهِ مِن دُونِ النَّاسِ فَتَمَنَّوُا الْمَوْتَ إِن كُنتُمْ صَادِقِينَ
Say (O Muhammad ﷺ): "O you Jews! If you pretend that you are friends of Allah, to the exclusion of (all) other mankind, then long for death if you are truthful."
Al-Jumu'ah:6Christians
يَا أَهْلَ الْكِتَابِ لَا تَغْلُوا فِي دِينِكُمْ وَلَا تَقُولُوا عَلَى اللَّـهِ إِلَّا الْحَقَّ إِنَّمَا الْمَسِيحُ عِيسَى ابْنُ مَرْيَمَ رَسُولُ اللَّـهِ وَكَلِمَتُهُ أَلْقَاهَا إِلَىٰ مَرْيَمَ وَرُوحٌ مِّنْهُ فَآمِنُوا بِاللَّـهِ وَرُسُلِهِ وَلَا تَقُولُوا ثَلَاثَةٌ
O People of the Scripture, do not commit excess in your religion or say about Allah except the truth. The Messiah, Jesus, the son of Mary, was but a messenger of Allah and His word which He directed to Mary and a soul [created at a command] from Him. So believe in Allah and His messengers. And do not say, "Three"
An-Nisaa:171Polytheists
وَلَئِن سَأَلْتَهُم مَّنْ خَلَقَ السَّمَاوَاتِ وَالْأَرْضَ لَيَقُولُنَّ اللَّـهُ قُلْ أَفَرَأَيْتُم مَّا تَدْعُونَ مِن دُونِ اللَّـهِ إِنْ أَرَادَنِيَ اللَّـهُ بِضُرٍّ هَلْ هُنَّ كَاشِفَاتُ ضُرِّهِ أَوْ أَرَادَنِي بِرَحْمَةٍ هَلْ هُنَّ مُمْسِكَاتُ رَحْمَتِهِ قُلْ حَسْبِيَ اللَّـهُ عَلَيْهِ يَتَوَكَّلُ الْمُتَوَكِّلُونَ
And verily, if you ask them: "Who created the heavens and the earth?" Surely, they will say: "Allah (has created them)." Say: "Tell me then, the things that you invoke besides Allah, if Allah intended some harm for me, could they remove His harm, or if He (Allah) intended some mercy for me, could they withhold His Mercy?" Say: "Sufficient for me is Allah; in Him those who trust (i.e. believers) must put their trust."
Believers
يَا أَيُّهَا الَّذِينَ آمَنُوا اسْتَعِينُوا بِالصَّبْرِ وَالصَّلَاةِ إِنَّ اللَّـهَ مَعَ الصَّابِرِينَ
O you who believe! Seek help in patience and As-Salat (the prayer). Truly! Allah is with As-Sabirin (the patient ones).
Al-Baqarah:153
Which brings about an interesting question:
Do any Qur'anic verses address atheists - those who deny the existence of a Divine Being?