The only Chocobo that has the ability to fly in Final Fantasy 9 is the Golden Chocobo. To upgrade your Chocobo to the Golden Chocobo, you must unearth several Chocographs from Chocobo’s Forest, and then locate them on the World Map.
Once you’ve received the Golden Chocobo, go to any forest on the world map and press the “Confirm” button while on your Chocobo. The Chocobo will begin flying.
Similarly, you must also land in a forest by pressing the “Confirm” button to stop flying.