Hydaelyn at large (is there a compilation of info yet?)

[Enkidoh]

I'm not sure if chocobos are native to Coerthas, but it's mentioned repeatedly (both in 1.0 and in ARR), that Ishgard has a reputation for being the elite when it comes to chocobo breeding and raising. Back when 1.0 was first released chocobos were not available to players and the in-game 'reason' given as to why npc handlers wouldn't let players ride them is because there was a disease affecting chocobos at the time called 'chocobo blight', and hence the Ishgardian stables had quarantined their birds to stop it rendering the species extinct.

When chocobos were added in the 1.19 patch though the npcs mentioned that the quarantine seemed to have worked and the blight had diminished, thus the Ishgardian stables released their birds to the handlers and adventurers were finally allowed to ride them.

Again this doesn't prove that chocobos are native to Coerthas, but at the very least the Ishgardians over time developed a tradition of chocobo breeding, which you would think would go 'hand in hand.'