The minimum lvl requirement to obtain this quest is 20
You must also be a part of the Grand Company. You will get the Chocobo Quest around 2 quests into the Grand Company quest line
The Chocobo mission asks you for a serpent chocobo insurance. You will need 1,000 Grand Company Seals for this, which you can get by farming FATES
You can then turn in those Grand Company Seals to the quartermaster for the license right next to the guy who gave you the quest.