A carbide drill or sacrificial spire point steel drill should work OK on mineral loaded plastics in hidden areas. I'd aim for a very light cutting pressure. I am extrapolating from Corian experience, so practicing on a junk piece of quartz counter material would be a good idea. If there are significant sized stone pieces embedded in the plastic, then diamond drills will be appropriate. Starrett may be a source.

In any case, JuneKnow's comment above presents a higher priority that may make the drilling moot.

We are the GC. In all the research I did this is the first I’ve heard about not placing quartz behind an oven. Not one of the professionals we hired mentioned this as an issue either. We have an electric slide in range with a pretty substantial back panel above the cooktop. I’ve been using the oven without any visible damage to the quartz, and it’s up now. I do see this has come up on the message boards before.

Thanks for your advice!