Quote has the clear upper hand here, and it'll take some specific preparations on the Chief's part to turn the tables.
The weapons each party uses are going to make all the difference in this fight. Master Chief has some advantages over Quote: he has Cortana, an advanced AI, to calculate trajectories, odds, and that sort of thing. He has better close-quarters combat training than Quote. He is faster and stronger, and he has a huge pool of equipment to choose from.
Meanwhile, Quote has five weapons in his arsenal, which means he can already fight effectively at any range, he has the Booster, which means he can avoid hand-to-hand combat, and he's got a life pot, which means he will take a very long time to kill.
I don't know how Cortana can help in this fight, and Chief's strength, speed and training are nullified by an opponent that can fly and shoot through walls. That means the only way to make this a fair fight is Chief's equipment. It has to be two things he can carry, and an armour ability. Let's assume the space marine gets some prior knowledge of Quote, otherwise he'll bring whatever he usually carries and get destroyed.
Chief's greatest asset would be stealth. If he tries to beat Quote at his own game, he will lose. If he could somehow dodge all the super missiles, he'd still have to fight through a cloud of lethal bubbles and a possessed (?) sword. So Active Camo would be the prime choice of armour ability, right?
WRONG, idiot. Everyone who's played a Halo match knows that you can see it. Quote's not going to miss the wiggly waves. Chief needs the
Hologram tech. Two seconds of misdirection would be enough for Quote to turn away from Master Chief's hiding spot and fire his cloud of bubbles, if he's got one. A plasma sword would most likely finish the fight at this point.
I think Master Chief's only other chance of winning is to use the Power Drain equipment. It's not an armour ability or a weapon, so I guess it's not allowed. Besides that, Quote wins by having more firepower and tougher constitution.