Note that Noah's wife is only mentioned in this unique verse in the Quran, while we can read a lot about Noah's effort for da'wah in the surah Nuh, we can also read about his son -who was a disbeliever- and didn't go on board of the arch (long parts of surat Hud are mentioning the story of Noah).
So what do mufasirin say about this verse?
This is taken from tafsir ibn Kathir
- "God sets forth an example to those who disbelieve:*
meaning, the disbelievers who live together in this life with Muslims, their mixing and mingling with Muslims will not help the disbelievers, nor will it avail them with Allah, until and unless they gain faith in their hearts.
- the wife of Noah and the wife of Lot, they were both under two of Our righteous servants,
means, they were the wives of two of Allah's Messengers and were their companions by day and night, eating with them and sleeping with them, as much as any marriage contains of interaction between spouses.
- but they acted treacherously towards (both betrayed) them
meaning, in the faith, they did not adhere to the faith sent through their husbands nor accepted their message. Therefore, all the intimate knowledge of their husbands neither helped them nor prevented punishment, hence Allah's statement,
- so they availed them naught against God,
means, because their wives were disbelievers,
meaning, to these wives,
- Enter both the fire with those who enter."
The part of the Ayah that reads, فَخَانَتَاهُمَا (but they both betrayed them) does not pertain to committing illegal sexual intercourse, but to refusing to accept the religion. Surely, the wives of the Prophets were immune from committing illegal sexual intercourse on account of the honor that Allah has granted His Prophets, as we explained in Surah An-Nur.
Al-`Awfi reported from Ibn Abbas,
"They betrayed them by not following their religion. The wife of Prophet Nuh used to expose his secrets, informing his oppressive people whenever any person embraced the faith with Nuh. As for the wife of Prophet Lut, she used to inform the people of the city (Sodom), who committed the awful sexual act (sodomy), whenever a guest was entertained by her husband.''
Ad-Dahhak reported that Ibn Abbas said,
"No wife of a Prophet ever committed adultery and fornication. Rather, they betrayed them by refusing to follow their religion.''
Similar was said by Ikrimah, Sa`id bin Jubayr, Ad-Dahhak and others.
Ibn 'Ashour in his tafsir at-tahrir wa-tanwir says that this example is set by Allah as a warning for non-Beleivers to show them that they won't have any advocates by Allah no matter how near they are to Believers, so unless they believe they can't seek refuge or help from anybody but Allah. The example shows that this applies even to the wife's of the prophets and is also a sign for the wife's of our Messenger (Peace be upon him) whom have been mentioned at the beginning of the surah telling them don't be like the wife of Noah or the wife of Lut but do your best to be like the wife of Pharao and Mariam daughter of 'Imran (whom would be mentioned later).
The quran doesn't mention the story of Noah's wife in any other place. And based on the story narrated in the old Testament (Genesis) it seems that she has betrayed him after the Noachian flood and Noah didn't know. Ibn 'Ashur also referred to biblical quotes speaking about Noah's wife being on the arch (she was part of the people who were saved and safely left it when they went on board and left it) but the bible -apparently- didn't mention her later. So either she became a non-believer afterwards or Noah married an other wife later whom was not mentioned in the Torah.
All tafsir books i consulted seem in consensus that no wife of any Prophet betrayed him in the meaning of illegal sexual intercourse. And agree on explaining the deception by saying they were non-Believers. In the case of Noah's wife some say that she used to tell people (the non-Believers) that her husband is mad and if somebody believed and followed him she was informing his enemy's about them, so she didn't keep his secret (note the relation between this and the beginning of the surah, were we are told that a wife of our beloved Prophet (peace be upon him) has disclosed some secret).
And Allah knows best!