Jax's Skill Max Order: Q → W → E
Level 1: Jax will start E 9 times out of 10, and so you will conduct the Quinn Cheese level 1. If Jax instantly Es, play the dance game (walk away from him, then he will walk back and you will then walk towards him). You ideally want to be in AA range when his E expires so you can AA him after to consume the harrier proc, but you want to be out of his E range. His E range is 300, and your AA range is 525, so you want to be in that zone of 225 units where you're in AA range of him, but out of his E range. He has an indicator around him so you can see what his E range is.
Level 2: Level 2 onwards, never use your E first. You will play the range game with Jax by autoing him to death whenever you can, and this will tilt them to eventually jump on you with Q. When Jax Qs you, instantly vault his Q in mid air to cancel his leap. This is what makes this matchup unplayable for Jax because your E cancels his leap in mid air. Save your E specifically to cancel his Q when he uses it, and only use your E for that.
Level 3-5: Your blind is really good against Jax because he can't ward hop away if he's blinded. The only way Jax kills you is by Charging E then flashing onto you to secure the stun. This is why it is ideal if you trade flashes with Jax whenevr possible. If you and Jax both have flash down, Jax has no win-con because you can E his Q instantly and then auto him to death after that since he can't land his E on you without flash, ever.
If you see Jax charge his E, instantly Q him because your Q blinds him for 1.75 seconds, and his E charges for 2 seconds. If you blind him 0.25 seconds or later during his E animation, his E will wear off when the nearsight ends. This is useful because some Jax players charge E for a bit, and then try to Q you last second to make his E stun you when he lands. So if he ever charges E, just blind him right away and you will ruin his plan.
Level 6+: The matchup doesn't change at all. I forget that Jax even has an R when I vs him lol. The only fear you should have is dying or roaming when he's 6 because he can melt plates post 6 if he's alone against a turret. Roam if you see a good opportunity, but just be very careful to leave Jax alone top lane post 6.
Jax is an incredibly easy matchup.
The main point to note is that Quinn's E can cancel Jax's Q damage and it will interrupt his Leap from his Q.
To play this matchup, save your vault in lane only to counter Jax's engage.
Typically, Jax will active his E, Counter Strike momentarily before he will engage with his Q.
The timing is important, but essentially just be ready to vault Jax when he is in mid air.
By doing this, you should leap out before he could get his stun off, and if you can avoid being stunned by his counter strike, then the matchup is quite easy.
Keep in mind that Jax's Q is now on an incredibly low cooldown, so he can leap onto you before your vault will come back up.
If you use your vault at all against Jax, be mindful of your positioning and do not walk in leap range of jax if your vault is down.
If you are able to successfully dodge Jax's E stun, then you should aggressively auto attack him quite a bit because he will have no way to trade with you cancel his leap/counter strike.
Important interaction is that Jax CANNOT use his Q to hop onto wards OR enemy champions, including you, if he is blinded.
If Jax activates counter strike and you blind him, he is unable to leap onto you so long as you stay out of his vision range, so you can screw jax over super hard if you're able to do this :) If you have the opportunity to trade flash for flash with jax, do it becasue his only counterplay is E-Flashing you.
If he doesnt have flash, you can E his Q jump, and then run him down after because he has no way to ever land his E stun on you without flash.