Raising Your Sharptalon

If you are using Ultimate Battle-Training Stone on your pets while the quest is active, it may not auto-complete. Relogging/reloading, dismissing the pet or changing to an alt may not auto-complete either. Resolution is to abandon the quest and take it again - it will auto-complete instantly.

As of Patch 7.1 there are 4 new Falcosaur pets. Each of the pets is associated with a different world quest around the Broken Isles. These pets all provide a quest chain to obtain a different mount. These quest chains require a couple of weeks to complete as each subsequent quest becomes available from your summoned pet after the daily reset after completing the previous quest.

To get each pet, you must first kill the Matriarch in the world quest area, which will cause the NPC version of the pet to spawn. The next step is to go to the NPC and interact with it and feed it its favourite food. If you don't have the food, you won't obtain the pet. There are vendors selling the required foods at the closest flightpaths to each world quest for most of these so just remember to pick some up on your way there!

Important note on orphans: I've encountered issues with the orphans not spawning after killing the matriarch. This isn't affected by the matriarch's nest but it does seem to be affected by getting the last hit on the matriarch. If you can't see the orphan after killing the matriarch then you may need to wait for the respawn. To check if you have completed the requirements to see the orphan run the scripts below.

[npc=115784] - Snowfeather Swarm!
This pet can be picked up from [npc=115737] in Highmountain (/way Highmountain 32.5, 28.2) after killing the [npc=115673] (/way Highmountain 35.1, 21.9). The food it wants is [item=128839], which can be picked up from [npc=108504] (/way Highmountain 29.8, 40.0).
Summoning this pet offers you the quest Raising Your Snowfeather which starts the quest chain to obtain [item=137578].
Subsequent quests: The Smell of Humans (Alliance) or The Smell of Orcs (Horde) > Hunting Lesson: Northern Hawk Owls > Allies in Highmountain > Snowfeather Team Up > Deadly Prey > Hunting Lesson: Spiketail Beaver > The Unfriendly Faction > Snowfeather Bonding > Hunting Lesson: Mudshell Conch > Teamwork Lesson: Naraxas > Hunting Lesson: Coralback Fiddler > Teamwork Lesson: Ursoc > Snowfeather Team Rumble > Snowfeather Reunion
/script print(IsQuestFlaggedCompleted(44879))

[npc=115787] - Bloodgazer Swarm!
This pet can be picked up from [npc=115741] in Azsuna (/way Azsuna 35.5, 7.5) after killing the [npc=115674] (/way Azsuna 35.6, 5.7). The food it wants is [item=128843], which can be picked up from [npc=109372] (/way Azsuna 47.2, 41.4) - this one is a long way from the quest area so try to plan ahead or you'll be running a long way to find food and will have to kill the matriarch again.
Summoning this pet offers you the quest Raising Your Bloodgazer which starts the quest chain to obtain [item=137577].
Subsequent quests: The Smell of Draenei (Alliance) or The Smell of Blood Elves (Horde) > Hunting Lesson: Erudite Manafiend > Allies in Azsuna > Bloodgazer Team Up > Dangerous Prey > Hunting Lesson: Fledgling Kingfeather > The Unfavorable Faction > Bloodgazer Bonding > Hunting Lesson: Juvenile Scuttleback > Teamwork Lesson: Serpentrix > Hunting Lesson: Felspider > Teamwork Lesson: Skorpyron > Bloodgazer Team Rumble > Bloodgazer Reunion
/script print(IsQuestFlaggedCompleted(44882))

[npc=115785] - Direbeak Swarm!
This pet can be picked up from [npc=115742] in Stormheim (/way Stormheim 79.8, 69.6) - jump down from the rocks above to reach here - after killing [npc=115671] (/way Stormheim 79, 76). The food it wants is [item=128763], which can be picked up from [npc=98112] (/way Stormheim 71.6, 60.6).
Summoning this pet offers you the quest Raising Your Direbeak which starts the quest chain to obtain [item=137579].
Subsequent quests: The Smell of Dwarves (Alliance) or The Smell of Tauren (Horde) > Hunting Lesson: Rose Taipan > Allies in Stormheim > Direbeak Team Up > Dire Prey > Hunting Lesson: Tiny Apparition > The Unkindly Faction > Direbeak Bonding > Hunting Lesson: Golden Eaglet > Teamwork Lesson: Fenryr > Hunting Lesson: Stormstruck Beaver > Teamwork Lesson: Krosus > Direbeak Team Rumble > Direbeak Reunion
/script print(IsQuestFlaggedCompleted(44880))

[npc=115786] - Sharptalon Swarm!
This pet can be picked up from [npc=115740] in Val'Sharah (/way Val'Sharah 47.0, 10.5) after killing [npc=115672] (/way Val'Sharah 48.0, 9.8). This world quest is somewhat awkward to get to but I recommend coming in from Highmountain (glider from a high point in the north-west or water walking around from the coast). The food it wants is [item=128837], which can be picked up from [npc=96784] in Dalaran (/way Dalaran 52.6, 34.4) - there are a variety of vendors in Highmountain who also sell the item should you choose to get to the world quest from there.
Summoning this pet offers you the quest Raising Your Sharptalon which starts the quest chain to obtain [item=137580].
Subsequent quests: The Smell of Night Elves (Alliance) or The Smell of Undead (Horde) > Hunting Lesson: Spring Strider > Allies in Val'sharah > Sharptalon Team Up > Destructive Prey > Hunting Lesson: Terror Larva > The Uncongenial Faction > Sharptalon Bonding > Hunting Lesson: Auburn Ringtail > Teamwork Lesson: Dresaron > Hunting Lesson: Vale Flitter > Teamwork Lesson: Xavius > Sharptalon Team Rumble > Sharptalon Reunion
/script print(IsQuestFlaggedCompleted(44881))

Check out [guide=4788] by Purgatorywolf for an in depth guide which includes details on requirements and tips for completion of each quest for your mount.

This is the mount awarded for completing the last Quest: [item=137580]

Here's the complete Questline:

You get one Quest every day but the date I'm posting this(Patch 7.1 Build 22989) it is still possible to get the next quest on a different character. So say you complete the quest on your druid you can still get the next quest from the same pet on any of your other characters.

For the quest that require you to complete World Quests you need have to have your pet out(and in your team for pet battle WQs), for the quests that only require you to defeat certain pets you don't need your pet out, it only has to fight and survive.

Dungeon and Raid quests can be completed on any difficulty.

Raising Your Sharptalon <<<<<You are here!
The Smell of Undead or The Smell of Night Elves
Hunting Lesson: Spring Strider
Allies in Val'sharah
Sharptalon Team Up
Destructive Prey
Hunting Lesson: Terror Larva
The Uncongenial Faction
Sharptalon Bonding
Hunting Lesson: Auburn Ringtail
Teamwork Lesson: Dresaron
Hunting Lesson: Vale Flitter
Teamwork Lesson: Xavius
Sharptalon Team Rumble
Sharptalon Reunion