The hunt quest will be available once you proceed throughs the main story. You’ll need to arrive in Trench Town and speak with Grand Marshal Corrigan. Once you speak with him, the saloon will open. The saloon is located in the middle of town and is accessed via the white and red double doors. Speak with the bartender about his scars and it will open the quest line.
Each hunt will open by continuing the main quest line in their respective areas.
There are a total of 10.
Cold claw - Eagle Peak -Antenna Cable Car
Oreyard Queen - Wreckage Zone - Crevice
Big jaw - Quarry - Outpost Gate
Splittooth - Forest Enclave - Collapsed Arch Pass
Moldfang - The Stronghold - Whitewater Reservoir north of camp
Spinewretch - Ancient Ruins - Distorted Sculpture
Wendigo - The Gate
Hauras - The Dunes - Crimsonleaf Oasis
Sandshifter - Desolate Fort
Typhon - Canyon of the Grand Obelisk - Wrecked APC.