Normally you get directed to your class hall by a NPC upon entering Dalaran which starts it off. Chances are you may have already picked it up unknowingly (I’ve done it myself just to dismiss the NPC before).
You could head to the Trueshot Lodge, which is the Hunter’s class hall, and see if there are any quests there to pick up. I’d recommend turning on “Trivial Quests” from the mini-map filter as it will make spotting them a lot easier.
If I remember correctly, you need to unlock the broken shore, and possibly do some questing there, before the class hall mount quest unlocks.
Then you’ll find a letter in trueshot lodge, close to the mission table, that starts the quest.
If you already completed the class hall campaign, you should do all the Broken Shore chapters before getting to start the Hunter Class Mount quest chain, which i assume start with a NPC in Deliverance Point after you done all of the Broken Shore chapter quests.
I’m pretty sure that you need to do the whole broken shore, and then when you get 15/15 an NPC in broken shore will have a quest that starts the questline!
Have fun uwu
You need to finish the Hunter order hall story. Then finish the Broken Shore questline, go into the dungeon and raid, get the treasures, sentinax and all that.
Then you get a new follower, need to send him on a broken shore mission. After that you go to the Broken shore, hand it in and you can begin the quest for the mount once all is finished at the Broken Shore. The quest will be on your order hall after you finish everything there.