There's definitely a connection between the image on Squall's ring and the GF Griever; however, I'm contending that the connection is pretty simple:
The design on the ring is just a picture of a legendary creature, i.e. Griever. Griever's also on his gunblade, necklace, and who knows where else. A popular icon for engraving or carving, apparently. The image of Griever is a winged, fire-wreathed lion, similar enough to the GF. It's never stated that Squall bought the ring, but then it'd be pretty weird if he found a mystical, magical ring and never bothered to look into its powers, then went ahead and bought a sword and necklace with exactly the same animal drawn on them.