You can read the story of Noah (AS) in these six chapters of Qur'an in detail:
And the most detailed one is the Hud chapter in which you can read about Noah (AS) in 25 verses ( 25 to 49 ).
In verse 11:40, Quran states:
حَتَّىٰ إِذَا جَاءَ أَمْرُنَا وَفَارَ التَّنُّورُ قُلْنَا احْمِلْ فِيهَا مِن كُلٍّ زَوْجَيْنِ اثْنَيْنِ وَأَهْلَكَ إِلَّا مَن سَبَقَ عَلَيْهِ الْقَوْلُ وَمَنْ آمَنَ ۚ وَمَا آمَنَ مَعَهُ إِلَّا قَلِيلٌ
[So it was], until when Our command came and the oven overflowed, We said, "Load upon the ship of each [creature] two mates and your family, except those about whom the word has preceded, and [include] whoever has believed." But none had believed with him, except a
so from the part "whoever has believed" we can understand that some other people were also in Noah's(AS) ship and they did survived, so one may think even if there were no people living in other parts of the world, we may be descendents of those other believers.
But in verses 37:75 to 37:82, Quran explicitly states that all of us are descendents of Noah(AS) :
وَلَقَدْ نَادَانَا نُوحٌ فَلَنِعْمَ الْمُجِيبُونَ
And Noah had certainly called Us, and [We are] the best of responders.
وَنَجَّيْنَاهُ وَأَهْلَهُ مِنَ الْكَرْبِ الْعَظِيمِ
And We saved him and his family from the great affliction.
وَجَعَلْنَا ذُرِّيَّتَهُ هُمُ الْبَاقِين
And We made his descendants those remaining [on the earth]
وَتَرَكْنَا عَلَيْهِ فِي الْآخِرِينَ
And left for him [favorable mention] among later generations:
سَلَامٌ عَلَىٰ نُوحٍ فِي الْعَالَمِين
"Peace upon Noah among the worlds."
إِنَّا كَذَٰلِكَ نَجْزِي الْمُحْسِنِينَ
Indeed, We thus reward the doers of good.
إِنَّهُ مِنْ عِبَادِنَا الْمُؤْمِنِينَ
Indeed, he was of Our believing servants.
ثُمَّ أَغْرَقْنَا الْآخَرِينَ
Then We drowned the disbelievers.
And about Noah's(AS) son, in verses 11:42 and 11:43, Qur'an proves his drowning:
وَهِيَ تَجْرِي بِهِمْ فِي مَوْجٍ كَالْجِبَالِ وَنَادَىٰ نُوحٌ ابْنَهُ وَكَانَ فِي مَعْزِلٍ يَا بُنَيَّ ارْكَب مَّعَنَا وَلَا تَكُن مَّعَ الْكَافِرِينَ
And it sailed with them through waves like mountains, and Noah called to his son who was apart [from them], "O my son, come aboard with us and be not with the disbelievers."
قَالَ سَآوِي إِلَىٰ جَبَلٍ يَعْصِمُنِي مِنَ الْمَاءِ ۚ قَالَ لَا عَاصِمَ الْيَوْمَ مِنْ أَمْرِ اللَّـهِ إِلَّا مَن رَّحِمَ ۚ وَحَالَ بَيْنَهُمَا الْمَوْجُ فَكَانَ مِنَ الْمُغْرَقِين
[But] he said, "I will take refuge on a mountain to protect me from the water." [Noah] said, "There is no protector today from the decree of Allah , except for whom He gives mercy." And the waves came between them, and he was among the drowned.