No you can't, at least at launch. Yoshi-P said they were "looking into it" for Goobbue and Coeurl as battle companion.
I honestly wouldn't want it to take away from more endgame raids, low man content, additional FATEs, and other such things to do at level cap but I'd like it. Allowing a mount to become a battle buddy would give people more drive to collect mounts beyond the usual "I have all mounts!" It allows the mount to serve a dual purpose and basically allows the developer to get "more bang for the buck" when it comes to development time. Edited, Jun 26th 2013 12:13pm by Viertel