Oh, I think you might need to speak to a bird that squawks on a box in front of the big house on Skull Island. If you talk to him about pets, you might very well find yourself with an ostrich after you do his quest.
Once you have the pink ostrich, morph it with anything. You will either receive the other pet you morphed with, or a blue ostrich. The only pink ostrich you can ever have is the one you receive from the quest Decius referred to. Any other ostrich you receive will be the blue one.
Good luck with training your little critter up! Maybe use some of those snacks from the vendors behind the morphing tent ? The Ostrich loves Marleybone Side-dishes and likes either Marleybone Origin snacks. It likes Side-dish snacks as well.