In order to access the Chocobo Air Garden, first you have to find its shadow on the World Map. It appears as a small dark dot, although it does not remain in the same place for very long. The two locations I remember are northeast of Alexandria and in the ocean, right in the middle of the world map, but there are more, although I don't think they're important as if you wait at one of these locations long enough, the shadow will appear. Fly over the shadow, use a dead pepper and there you are.