Is it permissible for a non muslim woman to read the Holy Quran in bed while lying down (for the sake of gaining knowledge of Islam)?

There are two aspects to your query: a) Recitation of the noble Qur'aan, b) In the context of a non-Muslim.

The Qur'aan and Ahaadith are replete with orders to respect the noble Qur' aan. Therefore, in principle, lying down and reciting the noble Qur'aan is disrespectful. However, in the case of a non-Muslim, the strict rules of respecting the Qur'aan are relaxed. It suffices that he/she (non-Muslim) approaches the Qur'aan with the intention and hope to learn about the Qur' aan. Imaam Abu Hanifa (RA) said, 'I do not have any objection in giving the Qur'aan to a non-Muslim as it is hoped he/she will be guided (by reading the Qur'aan).' (al-Ashbaah)

and Allah Ta'ala Knows Best

Mufti Ebrahim Desai