To what extent does ‘Sir Gawain and the Green Knight’ conform to the conventions of quest narrative?
Sir Gawain and the Green Knight conforms quite closely to the conventions of a classic quest. What are those conventions, and how does the story follow them?
First, in a conventional quest, a character (a character who will eventually be the hero of the story) hears a call to action and feels compelled to leave his current surroundings in pursuit of something. In this case, that hero is Sir Gawain. He's attending a New Year’s Eve feast at King Arthur’s court when an unexpected visitor, the Green Knight, arrives at the party. The Green Knight challenges someone in the party to join him in a game, then he mocks King Arthur. Sir Gawain takes this as his call to action. He doesn't want to participate in the game with the Green Knight, but he accepts this duty.
Second, in any quest, the hero faces trials and tribulations along his journey. Examples of Sir Gawain's challenges include his interactions with Lady Bertilak, who tries to seduce him. Gawain is terrified of offending her husband, who has offered him accommodation during the journey. But he must fend off Lady Bertilak's advances without offending her, too.
A classic quest also includes kind souls along the way who help the hero on his mission. In this case of that story, one of those figures is Bertilak de Hautdesert, the lord of a castle that Sir Gawain stumbles upon, and where he ends up staying before he encounters the Green Knight again.
A quest always includes fighting an enemy. Here, that's represented in the axe battle with the Green Knight: the "game" that the Green Knight initially proposed to the court at King Arthur's party.
Finally, a conventional quest typically ends in the hero achieving his aim, and if he doesn't, at least learning something about himself in the process. This particular story ends in an unusual way: Sir Gawain learns that the Green Knight invented the whole situation (including, confusingly, using magic to transform himself into Bertilak, Gawain's friendly host) as a way of testing Gawain's nerve. So although Sir Gawain doesn't exactly defeat his opponent, he does return to his home honorably.