Yes, it is permissible to pray on one’s bed so long as it is firm and the worshipper is able to place his forehead and nose firmly when prostrating.
Among the comments made by scholars about praying on one’s bed are the following:
1 – al-Hattaab said in Mawaahib al-Jaleel (1/520): It says in al-Tawdeeh: As for praying on one’s bed, there is no dispute that it is permissible, as is stated in al-Bayaan. End quote.
2 – al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’ (3/221): The condition for obligatory prayer is that the worshipper be steady in facing the qiblah throughout the prayer. If he faces the qiblah and does all the pillars (essential parts) of prayer in a howdah or on a bed or the like on the back of a riding-animal that is standing, there are two views as to whether his obligatory prayer is valid; the more correct view is that it is valid, and this was stated by the majority, because that is like praying on board a ship. End quote.
3 – Shaykh Saalih al-Fawzaan (may Allaah preserve him) was asked: Is it permissible to pray in a raised place such as on a bed and the like if one is not sure whether the ground is clean and he has no excuse such as sickness and the like?
He replied: There is nothing wrong with praying on something that is raised such as a bed and the like, if it is clean and is firm and will not shake or wobble and distract the worshipper. End quote from al-Muntaqa (2/143).
And Allaah knows best.