
Originally Posted by
Obsidia
All mounts cannot fly haha. Although we're at a point now of flying dogs, so I don't see why the 2.x extreme primal ponies can't lol. Anyway! There are no quests that 'unlock' flying mounts. Either you have mounts that fly after getting all aether currents in an area or you don't.
Not entirely true - to make regular chocobos fly you need to do a quick quest at level 52 from Ishgard Foundation called 'I Believe I Can Fly' after you have unlocked flying in at least one area (gained all it's aether currents). This will make your regular Company Chocobo, as well the Legacy Chocobo, Fat Chocobo, Original Fat Chocobo, Parade Chocobo, Draught Chocobo, Amber Draught Chocobo and Ceremony Chocobo all capable of flight. Other flying mounts like the Griffin can fly automatically as soon as you unlock flight in a given region and do not require a quest to unlock them.

Originally Posted by
Obsidia
As mentioned earlier, there's Black Chocobo, you're regular chocobo after completing a side quest and the Magitek Armor from completing the main story of ARR at level 50. There's another one you get after a certain point in the Stormblood MSQ too. Flying Mounts beyond these are optional and can be obtained in a lot of different ways. Buying Collector's Edition of Heavensward/Stormblood, paying cash in the Mog Shop for certain ones, Veteran Rewards, and rare drops from certain trials. If I saw a list of all your mounts I could just tell you which ones flew and didn't haha.
You actually get three flying mounts during Heavensward's main scenario (the aforementioned Black Chocobo, plus
the manacutter during the Aery quest and Midgardsormr in Azys Lla right at the end)
.
To determine which mounts are capable of flying and what can just walk or run on the ground (and fore that matter, which ones can swim and dive in Stormblood areas), open the Mount Guide and in the description for each mount, there is a special symbol under the name: a chocobo head means it can only run on the ground, where as if it has a wing symbol then it can fly.