Jayce's Skill Max Order: Q → W → E
Matchup Tip: Quinn E can cancel Jayce's melee form Q in mid air, which screws him over because he can't swap back for 6 seconds and he loses his gap closer when you do this. Cancel his Q, then auto him to death for 6 seconds.
Level 1: Quinn wins level 1 against Jayce. I like to Quinn Cheese level 1, and then instantly run away before Jayce can Q retaliate. Get prio in this lane because he can fight you for level 2 if you let him, but if you aim for 2 you will get it first.
Level 2: Jayce can E-Q you, which is a reliable poke that hurts if he lands it. Stand behind minions so he can't do this.
Level 3-5: Play aggro in this matchup. If Jayce misses E-Q, throw Q on him and get free trades. Ideally, save E to cancel his melee form Q, but typically jayce will only let you do this once. When you cancel his melee Q with your E, he likely won't ever go in melee form again. If you want to use E aggressively, Do E-AA-Run away instantly when he doesn't expect it. It's best to wait for your fleet proc to be up so you have enough movement speed to out run his melee form retaliation.
Level 6+: Jayce's only chance in this lane is laning phase. The second you get 1-2 items you outscale him in 1v1s extremely hard. Ninja tabis are really good because he goes lethality, and if he can't 1 shot you he's useless against you.
Both you and Jayce are ranged ADs, but his attack range is 500 while yours is 525 so you have the advantage! I run fleet in this matchup because the sustain helps you against Jayce's poke.
The most important interaction is to save vault to counter Jayce's engage.
Jayce, after using his ranged-form combo, will swap to melee form and use his Q dash (To the Skies!) on you, and your vault actually cancels the damage and dash of this ability.
This is important because when jayce swaps to melee form, he cannot swap back for 6 seconds.
So, when he goes into melee and Q dashes onto you, if you can cancel this dash with vault, you now have a window of 6 seconds to bully jayce and combo him without him being able to retaliate.
Be careful though.
Jayce's E in melee form can cancel your vault, but it's actually quite difficult to do, and that's why I say only vault him when he Q dashes onto you so that he can't E you in time to cancel your vault.
Ideally, vault his Q ASAP so that he can't hit you with his E knock back.
Jayce's E range is 240, which is less than half of our auto attack range, so he can't do much at all if you cancel his Q.
In ranged form, stand behind minions so that he can't Q you, or more important, E-Q you through his Acceleration Gate.
Try to bait out his ranged Q by moving unpredictably through your wave and also away from minions.
Build ninja tabis in lane, and you just mitigate most of Jayce's Damage.
Also note that when Quinn completes 1-2 items, jayce is unable to 1v1 you for the rest of the game.
His ONLY edge is getting a lead in lane phase, but with tabis and 1-2 crit items, you can all-in jayce whenever you want and he simply cannot win.
I have yet to lose to jayce in lane ^-^ Incredibly easy matchup with practice.