Short answer
No. The Quran does not mention Aisha's age or name. Her age at marriage comes from hadith: in Sahih al-Bukhari and Sahih Muslim she says the nikah took place when she was six and she joined the Prophet's ﷺ household at nine. Most scholars accept these reports; a few modern writers have questioned the numbers.
The Hanafi scholars, with the other schools, accept the reports in Bukhari and Muslim. In Hanafi law a marriage may be contracted by a guardian before puberty, but the wife does not move to her husband's home until she is physically ready, and harm to her is forbidden.
The four schools agree on this.
Explanation
The Quran speaks of the Prophet's ﷺ wives as "mothers of the believers" (33:6), and the ayat revealed about the slander against Aisha (24:11 onward) defend her without naming her. It does not give her age at any point. So the answer to "where in the Quran" is: nowhere.
The figure comes from Aisha herself. In Sahih al-Bukhari she reports that the Prophet ﷺ married her when she was six and that she joined him when she was nine. These are among the most authentic collections, and the great majority of scholars, past and present, accept the report. Some modern writers argue from other dates in her life that she was older; most hadith scholars do not accept those arguments.
Context the scholars point to: marriage at a young age was normal across the world at that time, and there is no record of even his fiercest opponents in Makkah objecting to it. Aisha grew into one of the greatest scholars of Islam, narrating over two thousand hadith and teaching the Companions, and she spoke of her marriage with love. Islamic law also makes the wife's physical readiness and wellbeing the condition for living together, not a number.
Laws on the minimum age of marriage today are a separate question that scholars treat under public welfare. To learn about her life, see Aishah and the wives of the Prophet.
The evidence
Quran, Surah Al-Ahzaab 33:6
ٱلنَّبِىُّ أَوْلَىٰ بِٱلْمُؤْمِنِينَ مِنْ أَنفُسِهِمْ ۖ وَأَزْوَٰجُهُۥٓ أُمَّهَٰتُهُمْ ۗ وَأُو۟لُوا۟ ٱلْأَرْحَامِ بَعْضُهُمْ أَوْلَىٰ بِبَعْضٍۢ فِى كِتَٰبِ ٱللَّهِ مِنَ ٱلْمُؤْمِنِينَ وَٱلْمُهَٰجِرِينَ إِلَّآ أَن تَفْعَلُوٓا۟ إِلَىٰٓ أَوْلِيَآئِكُم مَّعْرُوفًۭا ۚ كَانَ ذَٰلِكَ فِى ٱلْكِتَٰبِ مَسْطُورًۭا ﴿٦﴾
The Prophet is more worthy of the believers than themselves, and his wives are [in the position of] their mothers. And those of [blood] relationship are more entitled [to inheritance] in the decree of Allah than the [other] believers and the emigrants, except that you may do to your close associates a kindness [through bequest]. That was in the Book inscribed.
The Quran honours the Prophet's wives without giving details such as their ages.
Sahih al-Bukhari 5133
Sahih... تزوجها وهي بنت ست سنين وأدخلت عليه وهي بنت تسع ...
"He married her when she was six years old, and she was taken to his house when she was nine."
The source of the age is this report in Bukhari, not the Quran.
Where the ruling is recorded
- al-Hidaya, Kitab al-Nikah, chapter on guardians and equality. A guardian may contract a marriage for a minor; living together depends on her being able to bear it.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.