The Unfriendly Faction

Sounds like you are trying to do the Player vs Player Pet Battles, when you should be doing the Player vs Player World Quests. Summon you Snowfeather Hatchling, open your world map in the broken isles, find a WQ with the cross swords (this indicates a PvP WQ), go complete quest, rinse and repeat twice.

Strangely, I completed this after only doing one pvp world quest.

Credits go to: [guide=4788]
Day 1: Raising Your SnowfeatherLevel to 25
Day 2: HORDE The Smell of Orcs: Take your hatchling to visit the inn at Orgrimmar.
Alliance The Smell of Humans: Take your hatchling to visit the inn at Stormwind
Day 3: Hunting Lesson: Northern Hawk Owls: Use your hatchling to defeat 3 Northern Hawk Owls in Highmountain
Day 4: Complete 5 World Quests in Highmountain with your hatchling out.
Day 5: Complete 3 Pet Battle WQs with the hatchling summoned out.
Day 6: Deadly Prey: Complete 1 Rare Elite World Quest with your Hatchling out.
Day 7: Hunting Lesson: Spiketail Beaver: Use your hatchling to defeat 3 Spiketail Beavers in Highmountain.
Day 8: The Unfriendly Faction: Complete 3 PVP World Quests with our hatchling out.
Day 9: Snowfeather Bonding: Buy a Tiny Pack from [npc=96999] for your hatchling.
Day 10: Hunting Lesson: Mudshell Conch: Defeat 3 Mudshell Conch in Highmountain with your hatchling & 2 flying pets.
Day 11: Teamwork Lesson: Naraxas: Defeat Naraxas in Neltharion's Lair with your hatchling.
Day 12: Hunting Lesson: Coralback Fiddler: Defeat 3 Coralback Fiddler in Highmountain with your hatchling and 2 Elemental pets.
Day 13: Teamwork Lesson: Ursoc: Defeat Ursoc in Emerald Nightmare with your hatchling. (Can be done in LFR)
Day 14: Snowfeather Team Rumble: Defeat 3 Pet Battle WQ with your hatchling and 2 Undead Pets.
Day 15: Snowfeather Reunion: Return to the location where you got the orphan, heal the Matriarch. Receive: [item=137578]