The quickest and cheapest way to get there is to simply run South East out of the tunnel from Brutus.
There is another Chocobo Spoor at (J-10), between the Ignis Eruca and the Martello. You can reach it with Conflux #03.
Trade the greens to the Spoor and it will tell you to wait at a distance. Run a short ways up the hill and a chocobo will spawn at the spoor.
It will start to run around, then stop, spin a couple times, then run to another location.
Approach when it has its back to you and click on it from behind to get the message, "You've caught yourself a chocobo!"
If you approach when it is facing you, it will be spooked and run off to a new location, and may randomly run deep into the Coeurls and disappear.
If this happens, you will need to obtain a new bunch of Gysahl Greens on your own to try again.
Return to Brutus to complete the quest and receive your reward(s).
NOTE: Only one person can do this quest at a time. If someone else is doing it, you must wait for them to complete it before trading your Gysahl Greens. You will be unable to trade the Gysahl Greens until 60 seconds have passed since the chocobo was last caught.
However, a full party with the quest active will all have it completed no matter who pops or catches the chocobo. They even get completion if they're standing next to Brutus when the chocobo is caught.