Quartz countertops can absolutely go with a stone backsplash. It really depends on the colors of the two materials in question as to how well they will coordinate. Also consider flooring, cabinets, paint, etc. You have a lot of options depending on what look you are going for.

please have a look at the pictures i posted. countertop mostly white.

Quartz countertops are a wonderful durable option for any kitchen. They can coordinate wonderfully with all kinds of natural solid surfaces. If you are using the stacked stone shown above maybe look for a quartz that is just off white or white with a very light fleck of a gray or warm fleck.

consider how you will clean that backsplash. there are a lot of non-boring tiles that you can more easily swipe a sponge over.