For those eager to check out something (anything!) like me, here are some Pet Battle quests from the Alpha. Keep in mind some might be works in progress, placeholders, and/or NYI. These might just be the tip of the iceberg too, and more could be on the way. At least, we can only hope. :P
From the list, you'll see that there are 7 daily quests (account-wide) and one weekly (also account-wide). All of these award x4 Pet Charms each.
It's probably safe to assume that the daily quests will be trainer battles. Perhaps one per Legion zone? Can't say for sure yet; the trainer icons haven't been added to the in-game map on Alpha yet.
As for the weekly quest... I'm wagering that it will be similar to the current weekly Pet Battle quest, Battle Pet Tamers: Warlords. I should mention that that weekly is actually still available in Legion, in new Dalaran no less. It's odd but I think it's just temporary. Maybe the new weekly will replace it as testing continues.
The other undefined quests are a little tougher to pinpoint, however one Wowhead comment does provide some insight.
Bonus Objective RP:
"Mayla Highmountain says: That elderhorn calf is in danger! You have to help it!"
According to the Wowhead comment, Wildlife Protection Force is a bonus objective in Highmountain. There's even video from the commenter of the objective -- completing it involves defeating a level 25 epic quality Hungry Icefang in a Pet Battle.
It has 2 additional (random wild pet) teammates. The Hungry Icefang is classified as a boss pet and has the appropriate 'takes reduced damage' buff.
What's interesting about this bonus objective Pet Battle is that the NPC is not always present, as mentioned in the Wowhead comment. Yesterday, I traveled to the very spot where it spawns (shown in the user's video) and found that it was not there too. I checked back throughout the day, still didn't find anything.
You would think that this would be discouraging, but I had a hunch that this NPC spawn might be a special one.
Judging from the video, the time of day that the poster found the Hungry Icefang was nighttime. Perhaps late night. So I decided to check at around 10 PM server time, and ta-da! The boss pet was there in all its fearsome glory. (Poor little moose calf, I'll save you one day soon!)
I can't 100% confirm that the Hungry Icefang will only spawn at night, but I plan on checking back every night to test the idea.
While I'd love to be able to test the actual battle itself, I still don't have a full team of 25 pets yet on Alpha. Leveling up a pet is currently a pain as there's a nasty bug where your client crashes after completing a battle. So... more waiting!
I'm pretty eager to complete this bonus objective though; I'd love to know what the Common Bag of Loot holds. It's the reward for all of the undefined Pet Batttle quests (likely all bonus objectives), but also for other quests not related to Pet Battles.
I have to wonder, is its loot table shared among all of the various types of quests? Can I get a pet-related reward from doing some miscellaneous quest elsewhere? Or will the bonus objective determine the loot? It could even just be a placeholder reward too.
So stuff is still up in the air, which is expected since we're still in Alpha. I can't wait to be able to copy over my account data from live and have a full army of level 25 pets to test everything out. But that might not be possible until much later on in testing. I'll just have to be content with figuring out the mystery of the Hungry Icefang spawning...for now. :P
Edit: Additional questions about Pet Battle bonus objectives - Repeatable? Once per day, per toon? Account-wide? Or only once per toon, ever?
Also, update on Hungry Icefang spawning - Checked back at 6pm server time and it was spawned. Going to check at an earlier time to pinpoint the exact time it spawns. When it despawns...testing that might be tougher. If it's only up at night, I imagine it will disappear (early?) morning. I...am not a morning person lol. :P
Edit #2: Checked at 6pm, 630pm, and 7pm server time. Hungry Icefang wasn't up, which basically tosses a rock at my theory that he has a set spawn time. So the only noticeable pattern is that he doesn't show up during the day, and he only shows up at night. When at night is still up in the air.
Edit #3: It's noon and the Hungry Icefang is up. WTFF. There goes the theory that it only spawns at night. At this point I have NO IDEA what the spawn mechanic for this boss pet/bonus objective is. Random is random?!?!