Best way to take them is to make an enclosed taming pen that you can lure and trap them into. Trying to tame it without one is a rough task. Because once they get to a certain amount of torpor, they fly away very fast. Making it impossible to catch up with them.
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.