With this quest, you can fight side-by-side with your company chocobo as your battle companion, and your chocobo can learn battle abilities as it levels up.
Requirements
This quest is available if you are level 30 and have completed My Little Chocobo and have used the Chocobo Whistle from that quest to unlock your chocobo as a mount.
Starting the Quest
Talk to Docette at the Camp Tranquil in the South Shroud to begin the quest.
Speak with Luquelot
Go north to the Central Shroud (or just teleport to Bentbranch Meadows) and go west from the Bentbranch Meadows Aetheryte to find Luquelot.
Obtain a Fresh Bunch of Gysahl Greens
Go north to the area marked on the map and check on any of the Gysahl Greens to take it.
Deliver Gysahl Greens to Luquelot
Go south to talk to Luquelot and hand over the gysahl greens.
Dispatch Brood Ziz with Your Companion
You should now have gysahl greens in your inventory. Use them to summon your chocobo. One use of gysahl greens will summon your companion for 30 minutes. You can use another bunch of gysahl greens to extend that time by 30 minutes, up to a maximum of 60 minutes.
You may wish to open the Companion window, which you can access from Main Menu > Character, and take a look at the different options available.
Optional: Purchase More Gysahl Greens
There is a Merchant & Mender north of the Bentbranch Meadows Aetheryte who sells more gysahl greens in the "Purchase Items" category. You can also purchase gysahl greens from various vendors around Eorzea. See Gysahl Greens in the official Eorzea Database for a full list of vendors who sell them.
Defeat Ziz
Go northwest from Bentbranch Meadows to the large tree root with the wooden platform at the base. Go up the tree root to reach the area marked on the map. There will be quest icons in the minimap showing where to find Brood Ziz. If you haven't used gysahl greens to summon your chocobo, be sure to do so. Then kill three Brood Ziz.
Report to Luquelot
After you have killed three Brood Ziz while your companion is summoned, go back to Bentbranch Meadows and talk to Luquelot to complete the quest.
Chocobo Saddlebag
Now that you have unlocked your chocobo companion's battle capability, you also have access to the Chocobo Saddlebag. You can access it from Main Menu > Character. It provides 70 additional inventory slots. You can optionally subscribe to the premium version of the Final Fantasy XIV Companion smartphone app to double your Chocobo Saddlebag capacity.