I would speak to your fabricator about the stains and having them removed before you did anything yourself. Unless you put them there? Then you will want to follow the manufacturers recommendation on a poultice.

Is it a stain or an unpolished area? From the photo, it looks like it's the same color as the quartz close to the camera, but it's not smooth so it's not reflecting light like the surface around it. If it was marble rather than quartz, I'd guess it was from etching.

It's really hard to see what it is, but it's not soap residue is it? I have quartz in bathroom and on kitchen countertop and the quartz in the bathroom, in particular get "splotches" from soap (hand soap). I wash the countertop with Dawn detergent/warm water and then thoroughly rinse it and dry with a towel. If any residue remains, I vigorously buff it with towel. It disappears.

After installation of the bathroom countertop. there was some weird residues on it. I called the installers and they came back cleaned it off with acetone. I made them do it so they were responsible (and for getting it clean). Good Luck!

Try cleaning it with rubbing alcohol. That was what my installers used to wipe it down after install.