Quartz doesn't have to require a battery. It can incorporate a very small photovoltaic panel that converts light into electricity, along with something like a capacitor or battery to store the power, and run the watch. Citizen's Eco Drive is by far the best known, but Seiko and Casio use this extensively as well. Outside Japan, it's much less common, but Junghans makes their own; Skagen and Bering, IIRC, have a few as well.
The major advantage: no battery change. Ergo, no need to crack the case open and possibly mess up the seal. No trip to the mall. They do die eventually; the panels lose efficiency, the capacitors lose the ability to hold charge. But Citizen claims, IIRC, the storage device still retains 80% of its original capacity after 20 years. Almost all solar watches will run for at least 6 months with no light at all, so there's still plenty of capacity left. I did have 2 Casios die last year...but each was at least 30 years old.