Add white distilled vinegar to water in a ratio of 1 to 4 -- one cup of vinegar for every four cups of water.
Wipe the surface of the quartz with the vinegar solution and a cotton rag. Wipe the surface dry with a clean towel. This removes any dirt on the surface that could cause micro scratches.
Apply a small amount of plastic polish to a microfiber rag. You can use automotive polish designed for plastic since the quartz in counters is mainly held together with plastic.
Wipe the surface of the stone in a circular motion. Apply more polish to the microfiber rag until you buff the entire surface of the countertop.
Allow the polish to dry on the surface and then buff the dried polish off the surface with a clean microfiber towel. This will leave a bright shine behind.