Dododex tells you the amount of traqs to knock the quetzal out. But it doesn't tell you that the bird's torpor recovers quickly and it will take more traqs the slower you hit it. To knock the queztal out with the least amount of arrows is to hit it quickly as it trying to fly away. Don't wait for it to slow down.
The fun way is to jump off and grapple to your argie and whistle it to attack and since argies are slow it won’t catch up in time so tranq till it goes out also bring parachutes and consider liking so more people will know
If you manage to knock a quetzal out, try starve taming. If you never tried starve taming it’s just leaving a Dino’s hunger go to half way or lower and then you start feeding. I’ve done this many times and it’s very useful taming trick.