Bungie released Hawkmoon recently but you may not see the Exotic Quest in Spider's Dialogue. This is because you need to complete some requirements first.
Important note: If you already acquired Hawkmoon and are looking to get the quest for its catalyst and random rolls, that guide is in another castle !
Hawkmoon quest is gained by talking to the Spider in Tangled Shore and picking it up from him but there are several prerequisites that you need before it can be obtained.
First, you will need Season of the Hunt Pass, not Beyond Light expansion. It's good for those who didn't buy BL but might be a headache for players who may have gone for the expansion but not the pass.
Second, you need to have unlocked both the Moon and Tangled Shore on your character. If you haven't unlocked the Moon, the quest will not appear. If you haven't unlocked the Shore, well, you can't get to the Spider and Crow in the first place. Furthermore, if you are on a character that was created after Beyond Light kicked off, you also have to complete the tutorial quest chain in Cosmodrome.
Third, you need to have Season of the Hunt artefact, which is acquired with the quest that sends you to listen to Osiris' transmission in Zavala's office. If you didn't pick up the artefact yet and can't acquire it on your character, check the other characters on your account. It's possible you took the mission on one of them but never completed it.
Finally, when you have the artefact, you will be able to start Trail of the Hunted quest on the Moon. Complete the mission and travel to the Tangled Shore. Speak with Spider and Crow as necessary in order to trigger your first Wrathborn hunt. When you set up the lure, go to the nearby Cryptolith which is marked by the quest, trigger the hunt and fight the boss.
Once you take the boss down to 30 per cent or so health, it will disappear. Follow the green trail it left behind until you reach the area where you enter the instanced version of the hunt, complete it and you are set.
You can now go back to the Spider and pick up the Hawkmoon quest from him. After that, you will need to find a bunch of feathers and do some more chores to complete the weapon.