Olaf's Skill Max Order: Q → E → W
Level 1: ConductQuinn Cheese level 1 from the bush and then auto two times to proc phase rush. He will throw Q, and then run away when he goes to pick up his Q. You hard win level 1 with E, so play aggressive and use your range advantage to bully him.
Level 2-5: Play these levels extremely aggressively because he loses all trades with your E up. Your goal is to jump on Olaf whenver you get the chance to, and then continue autoing him until he runs away. With Phase Rush and W, he can't ever catch up to you, even if he lands his axes on you because of your slow resistance that you get from PR. He can't play the game until level 6, so punish him for this. If Phase Rush and E is down, play a little bit more back because you need one of the two to prevent his trades if he tries to go onto you.
Level 6: To beat Olaf level 6, Q-AA-AA when he Rs to proc phase rush as quick as possible. Do not use your E when he has R because he can literally walk forward and won't be pushed back at all. The goal is to proc phase rush and use that speed + your W procs to run away/side step his Qs. The second his R duration ends, instantly vault him because now he will be knocked back and slowed from your E, which sets him up to lose the following fight when his R expires. PR is good during his R, E is good the second it ends after. If you need to E during his R because you're 100% dead if you don't use it, then wait for Olaf to go into melee range of you. Quinn's E does not give you distance if you vault from max range, but the closer he is to you, the further your E will push you back, so use E when he is right on top of you to give you a little dash.
Seems like an impossible matchup post 6, but I classify it as medium if you run phase rush.
First, make sure to side step Olaf's Qs in lane with your harrier procs. If he lands a Q on you and tries running you down, Auto E Auto combo him to proc phase rush and run away.
This keystone provides move speed along with 75% slow resistance, so if Olaf is chasing you down and lands Qs on you, phase rush counters the slow from his axes.
The matchup pre 6 is heavily quinn favoured, so side step his axes and then play the matchup as any other melee matchup, which is be aggressive when olaf misses Q.
He has no gap closers without it, so you can hard trade him.
If you are getting in an intense 1v1 with him, ignite kinda earlier to apply griev wounds to mitigate his W heal and conq heal, and also to proc nimbus cloak from your runes to grant you extra MS.
Now, post 6 Olaf DO NOT DO NOT DO NOT vault olaf while his R is active. If you are extremely low and have no choice, then that is the only time you should vault him when his R is active. The reason you don't is becasue R grants him slow immunity, and so your E will not slow NOR knock olaf back, and that is a critical part about Quinn's vault ability.
How to win post 6 olaf is, when he activates R, instantly proc phase rush by either auto auto auto, OR Auto - Q - Auto. By procing PR, you have slow resistance and insane MS to make his Qs unable to slow you, and enough move speed to out run olaf in his R ability.
Side step the axes if you can, and focus on trying to just run away by procing harriers from passive or q when you can.
NOW, when his 6 second R ends, IMMEDIATELY use vault on olaf, and then re-engage on him because he will have no gap closer without his ult. :)
I rush bork here, so you should also save bork active for after his R expires to apply a slow on him. Practice makes perfect for this matchup, but post 6 is actually not bad with phase rush! Just save that E boys until his R expires always.