Just build a 2x4 or 2x6 pen with a dino gate on one side, placing stone pillars on the outer edges all along it. Make it near the green obelisk! Then fly the griffin into it thru the door with a pteranodon, fly out thru other end thru the pillars, land your mount, run to the other side, close the door, boom. Trapped griffin. Then tame with roughly 150 tranq arrows. Use mutton or prime meat as they are easiest to come by.
Griffins are just one of the best flying mounts in the game they can do some crazy dive bomb move and it takes damage, you can even use weapons on the mount and you can tranquil enemies from them and they cannot get you