Travel to the Den of Haal'esh in Hellfire Peninsula and search the Kaliri Nests for a Female Kaliri Hatchling. Use the Empty Birdcage to capture it and bring it to Falconer Drenna Riverwind at Falcon Watch.
The Haal'eshi arakkoa have domesticated a beautiful bird species called the kaliri.
I've managed to capture a young male and they appear to be trainable. The females are much rarer, but if I had one I could attempt to breed them. She has to be young, though... a hatchling. An older wild bird wouldn't survive in captivity.
Take this cage and travel south, to the Den of Haal'esh. Search the nests for eggs that are ready to hatch. If you bring me back a female hatchling, I'll reward you handsomely.