Ikky

BEWARE: If you complete this quest while at the 1000 battle pet limit, the reward will be discarded!
(and no, it's not mailed to you)

Edit: I opened a ticket and the GMs mailed me the pet.

Ikky is a battle pet learned by doing a short quest chain in the Spires of Arak zone.

First you need to have completed The Kaliri Whisperer. When you turn that in, these become available:

Ikky's Egg - Help Ikky hatch from his egg
Baby Bird - Gather grubs to feed Ikky

After this quest Ikky flies away! ... thankfully the story isn't done.
Return to Veil Terokk and you will find [npc=85320] waiting for you with the final step:

Ikky - Ikky becomes your friend (and battle pet - [npc=86447])

Edit: Added a bit more detail.

Quest chain to get [npc=86447] battle pet:

1. Return to Veil Terokk http://www.youtube.com/watch?v=g7eyULtYCuE
2. Last of the Talonpriests http://www.youtube.com/watch?v=y0OXG4w4T4Y
3. Ikky's Egg http://www.youtube.com/watch?v=PKEOANcU92M
4. Baby Bird http://www.youtube.com/watch?v=NmdxXNx1obo
5. Ikky http://www.youtube.com/watch?v=XiRuhai1Gu8