It was "Valdosta's Most Modern Restaurant," and you can certainly see why: the rocket-gantry bracings served absolutely no function, but made it look like a space port.
A good time to be a kid. Grown-ups might not have noticed that they were living in the future right now, but kids did.
No apparent connection to green turtles was offered, and no patron questioned the name. The sign no doubt had lights that went DOWN to the place, so you knew it was on the street instead of hovering 50 feet up in the air.