For deeper scratches, you may need to use a stronger abrasive, such as sandpaper. In all honesty, we wouldn't recommend that you go down this route. If the scratches are deep, it's best to call a natural stone professional in your area, who will have the tools necessary to remove deeper scratches and stubborn stains.
Once the scratch is gone, be sure to protect your countertops by using placemats a cutting board
and trivets. With a little care, your quartz countertops will look beautiful for years to come.