Praise be to Allah.
There is nothing wrong with reading the Quran from the Mus-haf whilst lying down.
Al-Bukhari (7549) and Muslim (301) narrated that the Prophet (peace and blessings of Allaah be upon him) used to lie in ‘Aishah’s lap and recite Quran.
Al-Nawawi (may Allaah have mercy on him) said in Sharh Sahهh Muslim:
“This indicates that it is permissible to read Quran when reclining and lying down.”
Shaykh Salih al-Fawzan said:
“There is nothing wrong with reading the Quran when lying down, whether one is lying on a bed or on the ground. There is nothing wrong with that. A person may recite Quran whatever position he is in, standing or sitting or reclining, and whether he has wudu or is in a state of minor impurity, if he is reciting from memory. But if he is reading from the Mus-haf, then it is not permissible for the one who does not have wudu to touch the Mus-haf unless he has done wudu.” (Al-Muntaqa min Fatawa al-Fawzan)
And Allah knows best.