Extremely difficult to tame. Best options are to bola it, and then use a longneck rifle with tranq darts. Otherwise trying to get it to fly into the side of a cliff and get stuff is the other option for a solo tame.
Wear ghillie suit before approaching. Crouch/crawl up to then when it lands and use tranquilizer darts instead of arrows. Arrows have a very high kill chance. Once it goes down circle the tame with spike walls as tapejara are very fragile!
After you manage to find one of these (usually high in the air, above quetzals. I've had the most luck around 50, 50) you'll want to make a trap. A Trapejara. Make a 3x3 cage out of doorways and put a ceiling on it.
Find your victim.
Fly level to it, using the back center doorway to line it up. Now for the chase, you'll want a quetzal with... a lot of movement speed. Mine is 211 and I struggle sometimes. Fly straight for the Tapejara, and once it clips into the structure, GET OFF OF THE QUETZAL. Your new buddy is trapped and afraid.
If you fail, let it fly out of render distance and try again. It won't be aggro'd anymore and it'll be flying normally do you have another shot.
If you just want to keep chasing it, wait about 15 seconds and it'll re-aggro, pausing for a second to do an animation. This is when I usually catch them.
The Tapejara is a versatile and highly maneuverable pterosaur known for its three-person saddle. While it may not have as much stamina and weight carrying capacity as other creatures, it has unique benefits in combat. The Tapejara can land on trees and other vertical surfaces, allowing both the pilot and the passengers to use weapons. Its rudder-like fin make it the only flying creature able to hover and strafe side-to-side.