Attach matching headboards or footboards to the two ends of a bed frame to make the sides of the couch. Footboards, being lower, are a more appropriate height for this purpose. If you have headboards, cut down the legs to make them lower or use them as-is for a couch with higher sides.
Place thick bolsters along one side of the mattress to form the backrest. The bolsters will remain in place without being attached if they are supported by a wall behind the couch. Because an unfolded twin mattress is wider than a couch, it is not comfortable to sit on without the bolsters. Make bolsters from blocks of foam or purchase them ready-made.
Cover the mattress and bolsters by draping drape a bedspread or blanket over the entire piece.
Add throw pillows for additional comfort and a more finished appearance.