Many massage parlours have rooms with comfortable thin Japanese tatamis or futons. It is not really a mattress but is at floor level. Just ask the MP for that.
For example Oasis on Van Horne has a large room with both massage table and tatami.
Royal massage at 5702 Sherbrooke West has a room with tatami and sometimes non-Asian masseuses. Better to phone in advance.
Aromass on Ct des Neiges and Queen Mary, Sassy on ct st luc rd., Aroma spa at 1007 Van Horne and 7ieme vie on Henri Bourassa also have tatami rooms.
Enjoy feeling grounded.