The best way in my opinion to catch a griffin is to yield a trap buil a. Structure large enough to fit the griffin. Use sliding doors. Hit the griffin then lead it inside the structure shut the doors and from there you can easily knock it out with tranc darts. But please do not use this method to Kill griffins.
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.