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
https://www.flickr.com/photos/438361...posted-public/
https://www.flickr.com/photos/438361...posted-public/

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.

Sonny