Is there a mount quest like this?

[Raldo]

The closest thing I can think of is Beastmen quests. Once you reach lv41, you can do the beastmen quests. Each of the ARR ones lets you purchase a mount for 120,000g once you've reached a certain level of reputation with them.

It's not exactly what you're asking for. They're solo quests (though you could help each other complete the quests), and they're not really something you can grind, since you're limited to picking up 3 quests per beastmen faction per day (up to a total of 12 per day). The quests are not hard at all, so I doubt you'll feel too accomplished when you finish them. You can probably finish 3 quests in 15 minutes or less. It just takes a couple months of doing them every day to accrue enough reputation, since the quests are so limited.



The only real grindy mount I can think of for you at your level are the mounts obtained by RECEIVING player commendations (those commendations you can award at the end of a duty). If you get 500, you get a Gold Magitek mount. If you get 3,000, you get a Fat Chocobo with a fancy hat. DPS don't really get commendations as much as healers though, so whichever one of you is the WHM is bound to get there first.


Let's see... other grindy mounts...

If you're not opposed to PvP, there are several mounts you get for winning enough of the group-PvP matches.

There's also some mounts you can buy in the Gold Saucer, once you grind enough MGP. Not really a group activity though.