The most likely issue, particularly for someone who used job and quest jumps, is that your Grand Company rank might not be high enough yet. You need to be at least 2nd Lieutenant before you can take the first hunt quest, Let the Hunt Begin at your company headquarters.

That one will unlock the level 50 daily/weekly hunts. It's also a prerequisite for the level 53 quest Let the Clan Hunt Begin that unlocks the first of the Heavensward hunts, with follow up quests at levels 56, 59, and 60 for the higher tier HW hunts. After all of those, and at level 61, the quest One-star Veteran Clan Hunt starts the Stormblood hunts.