Women can't recite the Qur'an or touch it during menstruation. What they can do is recite du'as and make adhkaar, regardless of whether they are from Qur'an or not. For example, reciting the du'a for traveling is permissible regardless of whether the woman is menstruating or not, despite this du'a being from the Qur'an. But, the woman cannot recite ayaat from the Qur'an.