A wife of Noah is mentioned in the Quran:
ضرب الله مثلا للذين كفروا امرأت نوح وامرأت لوط كانتا تحت عبدين من عبادنا صالحين فخانتاهما فلم يغنيا عنهما من الله شيئا وقيل ادخلا النار مع الداخلين
Allah presents an example of those who disbelieved: the wife of Noah and the wife of Lot. They were under two of Our righteous servants but betrayed them, so those prophets did not avail them from Allah at all, and it was said, "Enter the Fire with those who enter."
— Quran 66:10
This verse establishes that she was a disbeliever who betrayed Noah. The nature of the betrayal is not clear.
Some of the exegetes thought that she committed Zina and the person who drowned was her son and not the biological son of Noah. This is based on one interpretation of the following verse:
قال يا نوح إنه ليس من أهلك
He said, "O Noah, indeed he is not of your family;
— Quran 11:46 - See Tafsir al-Baghawy and Tafsir al-Qurtubi
However the stronger opinion is that he was his real son but not a believer. Both him and his mother were hypocrites who hid their disbelief. This is based on the fact that Allah calls him Noah's son:
ونادى نوح ابنه وكان في معزل يا بني اركب معنا ولا تكن مع الكافرين
Noah called to his son who was apart [from them], "O my son, come aboard with us and be not with the disbelievers."
— Quran 11:42
When accepting the stronger interpretation, a question might arise, why does Allah say: "indeed he is not of your family"?
There can be two answers:
Family (أهل) has been used in these verses in the meaning of follower in religion.
The verses use condensed speech: the word family (أهل) is used to refer to a specific portion of the family.
Allah promised to save the family of Noah with a certain exception:
وأهلك إلا من سبق عليه القول
your family, except those about whom the word has preceded
— Quran 11:40
Noah did not know that the son was among the exceptions since the son concealed his disbelief. Hence when he was drowning, Noah called upon Allah that his son was among his family who were meant to be saved:
ونادى نوح ربه فقال رب إن ابني من أهلي وإن وعدك الحق وأنت أحكم الحاكمين
And Noah called to his Lord and said, "My Lord, indeed my son is of my family; and indeed, Your promise is true; and You are the most just of judges!"
— Quran 11:45
Allah replied that the son is not among Noah's family. The meaning is that he is not among the family that was promised to be saved, but rather he is among the exceptions.
إنه ليس من أهلك إنه عمل غير صالح فلا تسألن ما ليس لك به علم
Indeed he is not of your family; indeed, he is [one whose] work was other than righteous, so ask Me not for that about which you have no knowledge.
— Quran 11:46