It is generally understood that women cannot read or touch the Qur’ān when they are passing through their menstrual cycle. People generally base their opinion on the following narrative reported by Ibn ‘Umar from the Prophet (sws):
لَا تَقْرَأْ الْحَائِضُ ولا الْجُنُبُ شيئا من الْقُرْآنِ
A lady who is passing through her menstrual cycle and a person who is in the state of janābah should not read anything from the Qur’ān.
As per the established Sunnah of the Prophet (sws), only three worship rituals are forbidden to women during this time: fasting, praying and offering the tawāf.
As far as the narrative quoted above is concerned, it is regarded as weak by authorities.
Reading the Qur’ān during this state whether by heart or through the Book is not forbidden by the sharī‘ah.
Of course, if as a matter of personal inclination, a woman does not feel up to reading the Qur’ān in this state, it is her choice but she should realize that she has not been stopped by the sharī‘ah in this regard.