How do I clean thinset from Quartizte Wall Tile?
If you are reasonably careful you can pick up a bottle of muriatic acid (the trade name for HCL) at the hardware store. Dilute it according to the instructions and gently scrub with a nylon brush. The acid will dissolve the mortar while leaving the quartz. 

It goes without saying that reasonable care includes not using excessive amounts or spilling it on any cement-based surfaces you're worried about etching. I'd suggest working from a pail with a brush so you don't apply too much. You'll see bubbles in the acid while it's reacting with the mortar which will slow and cease as the pH normalizes. Then you can wipe down the tile with a damp cloth and move onto the next.