Training chocobos in stables is one of the functions planned in the future Scenario Engine updates.
You can find the list here:
Planned Functions
If you ask about active leveling via FATE Farming, then no, it is not supported at the moment.
Our community assigned a low priority to FATE features due to inherent danger posed by open-world combat. Therefore we focus on implementing private instance activities and Assist Mode instead.
For additional information about open-world combat, please read this thread:
Asking the community: Thoughts on Open World Combat and User Protection