Your going to want your trap near the plains by green on, Have 5 Dino gateways lined up next to each other with 2 Dino gates on each end. Have each space big enough to just barely fit a ptera thru. After that have one gate opened and one closed and agro a griffin. Lead it thru the trap and fly thru the top then while it's still agro on the ptera on the closed gate side, close the other gate and tranq!
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.