Is there any verse in the Quran where it says a man or woman should marry only after reaching puberty?
" وَابْتَلُواْ الْيَتَامَى حَتَّى إِذَا بَلَغُواْ النِّكَاحَ فَإِنْ ءَانَسْتُم مِّنْهُمْ رُشْدًا فَادْفَعُواْ إِلَيْهِمْ أَمْوَالَهُمْ"
And test the orphans [in their abilities] until they reach marriageable age. Then if you perceive in them sound judgment, release their property to them. And do not consume it excessively and quickly, [anticipating] that they will grow up. And whoever, [when acting as guardian], is self-sufficient should refrain [from taking a fee]; and whoever is poor - let him take according to what is acceptable. Then when you release their property to them, bring witnesses upon them. And sufficient is Allah as Accountant.
Here, the word بلوغ النّكاح is used which translates to "Mature enough to be married". That is, they must have matured; i.e. mentally developed.
Maturity period starts after puberty, but its time may vary from person to person. For example a person may have reached the age of puberty, at the same time as “age of maturity” other person may have not reached the “age of maturity” although a few years had passed after puberty.
So, having attained maturity is compulsory for marriage besides having reached the puberty.