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.
It is permissible for a woman to recite the Qur'an without wearing the Hijab. She is asked to wear the Hijab only at the place where alien men can see her and then she should cover her whole body. Moreover, she is also required to abide by the Hijab while performing the prayers. In the prayers she has to cover whole body, save her face and hands (to wrists).
In addition, it is also permissible to read the Qur'an without having Wudu and also while she is menstruating or in post-childbirth bleeding if she does not touch the Qur'an directly. Touching the Qur'an is permissible only for those who are in the state of Taharah (Purification).
Allah knows best.