Several things…first, scratching can be caused by subsurface damage from coarser laps that has not been entirely removed by finer laps. Leave enough material to be cut down by a 600 or 1200 lap if you are using something coarser. Quartz is a material that is very susceptible to subsurface damage. Same with the 600, spend some time on the prepolish lap before polishing.
Secondly, the best polish for quartz is cerium, a good grade of cerium, not a cab grade…sometimes called French cerium or facet grade cerium.
Third, if you are using a slurry of cerium and water, don’t. This is the old method and it works, but the polish comes in just as the lap dries out and starts to tug. Then if you don’t add just a little spritz of water, your slurry dries out and you get scratches. It’s also possible to use to much cerium and get it to ball up and scratch. Best advice is to use a cerium coated or impregnated lap or a cerium crayon where the cerium is in wax. The wax prevents the scratching, as does the coated or impregnated lap. Use of a lucite lap is really old news for quartz…the coated or impregnated laps or the crayon on a suitable lap work much better. There are also composite surface laps that work well for quartz. Try looking at Gearloose’s products and Marsh Howard’s Lightning Lap products. Both will have recommendations for quartz. If you are using a BATT or a lucite lap, get the crayon (such as BATTstik). Hope this is helpful. Polishing with a modern method is much easier and faster. Quartz should not be a difficult material at all. It only has that reputation because folks don’t know how to approach it.