The way regular hunting quests unlock in this game need some work, I think. It's needlessly obtuse. And when you reach high rank, unless you pair up with players who do have the monster hunt unlocked, you just have to fuck around in an expedition (or another monster hunt) and try to accidentally encounter that monster.
For example, when you reach high rank, you get a delivery request for Jyura parts (to upgrade something in the town, I forgot what -- I also needed parts to upgrade my water lance too anyway). But the HR hunt for Jyura isn't available. So the only way to get those parts is to either pair up with a player who does have that hunt, or, if solo'ing, go on an expedition in the Wildspire Waste and hope to accidentally meet Jyura. And the monster doesn't always immediately spawn. When I tried to unlock HR Jyura, I went on that expedition, but the 3 monsters spawned in the Wildspire Waste were Barroth, Rathian and Kulu-Ya-Ku. I had to either kill one of those, or wait until one "left the locale", for Jyura to finally spawn, which, upon encountering it, finally unlocked the hunting quest (got the "a new optional quest is available" upon returning). I suppose I could have also hunted HR Barroth and hoped for Jyura to show up in a turf war or something, but yeah.
Unless I'm missing something, because I couldn't find a way to unlock the hunts otherwise. Why not just give the hunts immediately available at the quest counter? So weird.