When you get a new mount in this game, the item in your inventory is a 1 time use sort of thing that will add the mount to your mount guide. If your company chocobo is not in there then you need to do the quest for it.

In order to do the quest for it, simply follow your main scenerio and it will eventually lead up to the chocobo quest. You cannot simply get the quest as it's part of the long chain of main scenerio quests.