In order to start the Queen of the Hunt drop quest, you'll need to find the Hunter's Shack. This very small and unassuming wooden shack can be found in the northern part of The Mire (North East side of the map). It lies roughly in-between the Abandoned Shack and Southampton Estate. Please refer to the images below for reference:
The mission requires you to use the terminal inside the hut in order to initiate the quest. You will then be required to travel to 1 of 3 locations and kill a reletivaly high level creature (ie a Wendigo). I would suggest equipping a good weapon to deal damage with, as well as having plenty of stimpacks with you if you intend to solo this quest. You will then have to collect a sample from the dead creature then return to the hut to analyse said sample. Once complete, your achievement should pop.