Depends on what you mean by the Georgia mountains. From a Florida perspective?
Lake Sidney Lanier and Hartwell Lake are about the fall line, 600-700 feet. Allatoona Lake is at 850 feet, Carters Lake about 1100 feet. All have COE recreational facilities.
But is that in the mountains? Most of the state of Kansas is higher than 1100 feet, and nobody thinks they are in the mountains, even in the west where the elevation is over 3500, because it still looks flat. On the other hand, people will drive through hills and valleys, 300 feet to 1200 feet, on the Ozark Plateau, and convince themselves they are in the mountains because of the local relief.
So what do you seek, bumpy scenery or elevation? Elevation in Georgia is in the far northwest corner, edge of the tail of the Blue Ridge, where a few ridges go above 2500.
At almost 1800 feet elevation, Blue Ridge Lake and Nottely Lake might be considered mountain lakes, but there are no COE campgrounds. There is a RV park on Nottely Lake, outside Blairsville.
COE will have its campgrounds on flood control reservoirs it manages, on on navigable waters. You might check out Forest Service camping opportunities in the mountains.