Before you all go naming your griffins "Buckbeak" lemme lay some mythology on ya. Buckbeak from Harry Potter was actually a hippogriff which is a horse combined with a eagle. A griffin is a lion combined with an eagle. Just keep this in mind before going on a Buckbeak naming spree.
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.
These things are slightly terrifying. Taming is very similar to an argentavis, but since they're so fast they are hard to catch once they start to fly away. A taming pen is recommended unless you have a tapejara with another tribemate to shoot at it. Allosaurus kibble works the best but alas mutton is a wonderfulalternative since the ovis is abundant on Ragnarok. These creatures are a bit narcotic heavy so be ready! Very worth it though!
To tame this magestic beast, hit it to anger it, take it high into the sky (away from all things itll try to attack) and lure it into a trap made for pters to be able to pass through. They create dust which makes it hard to see where you need to hit at times. Its takes a ton of tranq darts. But its worth it. Pump stam, health and weight and its a faster tape. And its beautiful. Have fun :P
Build a pen, lure the Griffin into the pen, knock it out, let it wake up, knock it out again, then feed it mutton. 100% taming effectiveness. With an ascended crossbow it's not to many arrows for a knock out. Like 40 each time for a 150. Hope it helps!!!
As always, any carnivore, including this one, mutton is always the way to go. Kibble is for your breed imprints. Mutton always takes faster and is so close to kibble that it doesn't matter. So bahhh ram ewe
If you do not want to build a trap, try and knock it out in a thin canyon area, this way it is likely to get stuck on the canyon walls trying to escape, you will need high health / good armour for this.