Create a pattern for the laminate backsplash out of paper. Measure and cut the paper to size and then test it to make sure it fits.
Place the paper pattern over the 1/4-inch plywood panel. Trace the pattern with a pencil, and cut the plywood panel to size with a circular saw. Test the plywood to ensure that it fits in place.
Place the paper template for the backsplash over the laminate panel. Cut the laminate panel slightly larger (1/4 inch) than the paper template. Use laminate shears to cut the laminate.
Spread laminate adhesive on the back portion of the laminate panel and the front of the plywood substrate. Let the adhesive dry completely. Set wooden dowels on top of the plywood substrate. Place the laminate on top of the dowels with the glue side facing down. Carefully pull the dowels out from between the two layers one at a time. Push the laminate down firmly on the plywood. Trim the excess laminate with a router.
Test the fit of the laminate/plywood backsplash. If it fits, apply a generous amount of construction adhesive to the back of the plywood substrate. Push the plywood into place on the wall.
Finish the look of the backsplash as necessary with small wooden trim. Cut the trim on a miter saw, and nail it in place with finish nails.