It is not obligatory for a woman to put on hijab in order to read the Quran because there is no evidence to indicate that this is obligatory.

Hijab is obligatory for women. Allah commanded women to cover their adornments, including the face, bosom and all their adornments. Hijab has been enjoined upon women as a protection against abuse and temptation.

We cannot say for certain whether a woman who does not wear hijab will enter Hell, but she deserves the punishment of Allah because she has disobeyed His command to her. With regard to her specific fate, Allah knows best what it will be.