Yes, the 20 gauge is a better choice. Not that a 410 can't kill a quail, the pellets from each will have the same energy, but the 20 gauge just has a lot more pellets and is easier to hit with.
Rather than buying a cheap 410, take a look at used 20 gauge shotguns. I picked up a used Remington 1100 2-3/4" 20 gauge a few years ago for just about nothing. Most people want the 3" 20 gauge magnums, and all those perfectly good 2-3/4" 20 gauges are just sitting around gathering dust. Shells are cheap, and they are a pleasure to shoot.