A good webiste to track MSQ quests : https://ffxiv.consolegameswiki.com/w...cenario_Quests
--> find out what your current MSQ is, and find it on the list.
(Pay attetion to MSQ lvl, not your own level , as you often outlvl the MSQ)

How unlock mount
- Be level 20+
- joined a Grand Company (MSQ).
- Pick up the Blue quest;[ My Little Chocobo]

https://ffxiv.consolegameswiki.com/wiki/Chocobo.