Bloodgazer Hatchling is a new falcosaur pet added in patch 7.1. There are four types of pet in four areas. Different from the previous methods to catch pets, to let hatchling spawn respawn, you need to do World Quest: Sharptalon Swarm! And Direbeak Swarm! And kill Matriarch of the brood in the quest zone. The Hatchling spawn will respawn till you catch it or the quest reset. Each falcosaur has a special type of food that they respond to. Bring it the right food, and you'll get a wonderful reward! Yes, you will get the hatchling spawn.
Will it be over to collect the Hatchling? No, this is only the start. When you summon it out, the hatchling will have a quest for you. Level them up to 25 will complete the quest. The hatchling will offer you quests for you one by one, including pet battle in the wild, explore new maps, world quests, dungeons, and raid bosses killing. You should summon your pet beside you btw.
Bloodgazer Hatchling requires a nap time after completing a quest, and it will continue to provide you with new quests after rest.

Each Hatchling will offer an Ultimate Battle-Training Stone as a reward, but it is not the ultimate reward, because when comparing with the name of four new mounts added in 7.1, you'll find that the Bloodgazer Hatchling may grow to a relevant mount Predatory Bloodgazer.