2022: The easiest possible method to execute, but requires a bit of setup prior to it.

1. Any flyer will be be fine. I used a Wyvern, and a Sinomacrops.

2. Have parachutes at the ready.

3. Position yourself over the Quetzal you want to tame.

4. Drop on it, and while falling, hit it with a Net Projectile from the Harpoon Launcher. You can unlock both on Lost Island. It will fall to the ground, trapped.

5. Tranquilize while it’s down. You’ll have plenty of time.

6. Tame.