Easiest way to tame a quetz is to have a quetz with a cage built on it.(assuming you or our tribe already had one) Trap the wild quetz with the quetzal cage being sure to put down your walls at the front of the quetzal cage to lock it in. Hop off your quetz and shoot the recommended amount of arrows into it until it drops. Its at this point you have two options, you can either tame it in the air or have your tribe mates clear the ground area of any agressive dinos below it so that you can the drop the wild quetz on the ground where your not as restricted to what your able to do during the taming process. Be sure to put double spikes around the wild quetz on the ground because some dino such as dilos and carnos can attack through the spikes. This is the easiest tamin method to date once you already have a quetz established.
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 drop a quetzal due to knockout, and it lands next to a giga...just know, the quetzal belong to the giga. Now you get to watch your would be quetzal get eaten by a giga. Now you can go home and cry.