An upholsterer would have and can do what needs to be done at a cost.
I suggest you go to Hobby Lobby (Hob-nobby Lobby) or your local fabric store and get at least 2, maybe 3, ready made buttons with eyes on the back side.... like this
These buttons are about the size of a nickel. Quarter size buttons would do fine, also. Attach a string to the eye and run the string from inside the back of the cushion (along the zipper area). Attach the string to the plywood, pulling/securing the cushion to the back of the ply. Maybe get some extra buttons, in case a button or eye breaks, you'll have replacement buttons. This may be your lease expensive fix and easy to do. Hobby Lobby or fabric store may have a large needle for threading string through the fabric.