I have a piece of stained red oak plywood that is 1/4" thick and ~ 1 square foot in area. I will be gluing it to a 1" x 2" red oak board. Both the plywood and board have been stained and finished with polyurethane and there will be no weight attached to the glued on section of plywood. I'm using Titebond III Ultimate Wood Glue. Can I expect the glue to bond tight enough to hold the plywood to the board?
Bonding anything together without sanding either down to bare wood grain, or at least scuffing the finish surface prior to bonding, will be tenious at best. Having said that, if all you are trying to do is glue them together so one for not move, the bonded material is not weight bearing or subject to vibrations....it might work for a while.
Your choice of glue, in my opinion, is designed for glueing unfinished woods together (for the woodworking craft). Once the grain is sealed by the finish, a different type of bonding is required.
Construction adhesive or a dab of silicone might provide a better, longer lasting bond.