Quartz countertops can be damaged or discolored by certain chemicals that bleach or etch the resins used to make the countertop. Technically, such spots are not true stains where a substance has absorbed. Rather it is chemical damage to the countertop that is often permanent.
For this reason, it is best to use products made specifically for use on natural stone to avoid damage. Quartz countertops are 93% natural stone bound together by resins so it makes sense to care for it like granite or marble.
In other cases, a substance could cause an actual surface stain (vs. damage to the resins). This is especially true for white or light-colored quartz countertops. Things like tea and coffee spills are common causes.
You do have some options to try and remove or repair stubborn spots, stains or discolorations.
The first thing to try is scrubbing with Bar Keeper's Friend. This is sometimes successful removing various quartz countertop spots and discolorations. If this does not work, then contact the manufacturer for additional guidance.
At this point, professional repolishing may be your only potential option to restore the color and shine, but it may not be possible or successful. Unfortunately, in some instances, the damage will be permanent.