If you absolutely can't get to the mounting bolt, here is one way to fix it.
Plan on replacing the wood. If the wood has any rot or the hardware inside the cushion has rusted, you may be able to pull the cushion off by hand. If not don't try prying it off and causing other damage. Use a razor blade or knife and cut the vinyl around the edge of the cushion. Remove it and remove the foam. Now you can see how the bolt is held in the wood. It could have a bolt or it could be a stud screwed into a tee nut. There are several ways to remove the head of the bolt or tee nut. Drill out the bolt or use a cut off wheel on a Dremel tool. Once you remove what is holding the bolt to the wood, you can easily remove the piece of wood.
You will need 2 stainless screws about 1 1/4" long for mounting the cushion. Cut the new wood. Before covering, on each side of the wood find the middle of the sides. Drill a hole on each side about 1/4" from the edge. The hole size should match the size of the screws. Now you can cover the cushion. If you glue the foam to the wood, don't glue the side edges especially close to the holes. Once covered, you are going to cut a small slit in the vinyl with a razor blade. These cuts should be about 1/2" to 3/4" from the back of the cushion, in line with the holes in the wood. The cuts only need to be the width of the screw head, no larger. To mount the cushion, carefully insert the screws through the slits and into the holes in the wood. Run the screw in carefully so you don't mess up the slit in the vinyl. Tighten until the head of the screw disappears into the vinyl and tighten it down. The slits will not be noticeable since they are on the sides close to the rear of the cushion. If you're careful when installing the screws the slits will come together almost disappearing.
There are a lot of boats out there with the cushions mounted this way.