In the spirit of trying the cheapest, easiest thing first, you may want to try apple cider vinegar (not apple cider "flavored" vinegar, check the label). Spray it on until it puddles, then let it set for ten minutes before wiping it off.
i have been called to "fix" solid surface and quartz countertops that were "defective." The first time, just on a hunch, I brought a bottle of ACV. When i showed the store manager that it was scale buildup, and not an issue with the countertop, he asked if they could use Lime Away or CLR. I told him he could, if he wanted to pay $30 a gallon, but ACV is like $6.
I have never tried to dry polish quartz. I can't imagine it would go well, as the acrylic binder can heat up and discolor. Even the heat from dry cutting faucet holes on site can change the color of the edges of the hole, but it gets covered with a faucet.