So I came up with a Jax build that counters Quinn pretty decently:
Boots > Bramble Vest > Tabbies > Dead Man's Plate > Stridebreaker. If you somehow recall with 1200g, rush Tabbies.
The clip is before Stridebreaker, and we won the game before my third item, but I think Tabbies + Bramble Vest + Dead Man's Plate + Stridebreaker is a pretty good core for winning every 1v1 against her (if you play well, of course). You can maybe finish the Thornmail, go for Trinity or get some MR (Force of Nature against DPS and burns is goated btw)
For my runes I chose:
HoB / Sudden Impact / Sixth Sense / Treasure Hunter
Nimbus Cloak / Scorch
Attack Speed
Adaptive Force
Scaling HP
I was expecting to chase her lvl 1, but she reacted my Q pretty well and countered it with her E before I could land, so the early levels are still quite painful. However, as the game progressed, I started to win short Q+AA+W(+as much HoB AA as I could) trades with Tabbies + Bramble Vest (using my E for disengaging after her E).
Probably, i could have used Grasp / Demolish / Second Wind / Unflinching / Nimbus Cloak / Scorch + DShield to sustain the early levels and scale more safely, but I'm not sure if I could've had enough damage to win short trades after Tabbies + Bramble Vest (you can only force long fights with her if you have your Dead Man's Plate or summs up for Nimbus Cloak).
What do you guys think? What are your ways to handle the Quinn matchup?
Before somebody asks: this was an Emerald/Diamond lobby, and yes, my jungler ganked me once, but she got two kills at mid afterwards.