They make mounts for use on quartz countertops that are epoxied to the underside of the counter. I think it would be a good solution without risking running a screw through the top surface. They're little metal plates with machine threads, comes with machine screws. I just installed my own quartz and used them and they worked perfectly. Set of two was maybe $3. They were in the countertop area of the big home improvement store. You would also need a tube of epoxy.