Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
A woman is asked to acquire the knowledge just as a man is asked. The Hadith addresses a woman equally as a man. The Prophet (Sallallahu Alaihi wa Sallam) said: “If anyone travels on a road in search of knowledge, Allah will cause him to travel on one of the roads of Paradise”. The best knowledge is learning Qur'an and related sciences for the Hadith of Uthman that the Prophet (Sallallahu Alaihi wa Sallam) said: “The best of you is who learns Qur'an and teaches it to others”. [Al Bukhari].
Therefore, you may know that a woman’s learning Qur'an by heart is not merely permissible but it is among the best and excellent deeds.