It is not obligatory for a woman to put on hijaab in order to read Qur’aan, because there is no evidence to indicate that this is obligatory. 

Shaykh Ibn ‘Uthaymeen said that reciting Qur’aan does not require covering the head. 

Shaykh Ibn ‘Uthaymeen said in his comments on sujood al-tilaawah (prostration required when reading certain verses of the Qur’aan): “Sujood al-tilaawah is to be done when reading Qur’aan. It may be done in any state, even with the head uncovered, etc., because this prostration is not subject to the rulings on prayer.”

(Fataawa al-Jaami’ah li’l-mar’ah al-Muslimah, 1/249. See also Question no. 4908) 

“With regard to reading Qur’aan, it is not permissible for one who is in a state of janaabah (impurity following sexual activity) to read Qur’aan either from the Mus-haf or otherwise until he has done ghusl, because it was narrated that nothing prevented the Prophet (peace and blessings of Allaah be upon him) from reading Qur’aan except janaabah. But if a person has broken his wudoo’ (minor impurity), then it is permissible for him to recite Qur’aan from memory because of the general meaning of the evidence.”

(Majmoo’ Fataawa al-Shaykh Ibn Baaz, 10/150) 

It is better, and is mustahabb, to do wudoo’ before reciting, because this Qur’aan is the Word of Allaah. So it is better to do wudoo’ before reading it as a sign of respect. But touching the Mus-haf is not permissible unless one has done wudoo’, because Allaah says (interpretation of the meaning): 

“Which (that Book with Allaah) none can touch but the purified”

[al-Waaqi’ah 56:79]

(Majmoo’ Fataawa al-Shaykh Ibn Baaz, 10/150) 

And Allaah knows best. 

With regard to women who are menstruating or bleeding following childbirth, see Question no. 2564.