Neeko's Skill Max Order: Q → E → W
AD Neeko's Skill Max Order: W → E → Q
Level 1: Quinn Cheese level 1 and run her down. She will use her Q likely to retaliate, but you hard win it. Keep autoing her until she runs back to tower.
Level 2: Neeko will have E now, which roots you. Stay close range to neeko because her abilities are longer range than you. Always stand in E range to her if possible so that if she uses an ability, you can use E on her to juke her Q. If you want to use E aggressively, use E when she isn't expecting it, and then instantly side step after you E to juke her E. She will use E right after your E, but if you are quick enough you will bait this ability for her.
Level 3: Your harrier will mark the correct neeko if she is marked before she uses her W, so keep that in mind. If Neeko uses Q on wave, instantly E her for a trade to bait her W. If Neeko Ws, you have a 20 second window to punish her because Neeko is defenceless without her W.
Level 6: If Neeko Rs, E her quick -AA consume harrier, then run back with your amplified speed. She has to channel her R for 1.25 seconds, so you have a short window to launch your combo to escape before her R roots you.
Watch these Neeko Interactions to see combos for every ability and how to counter them as Quinn.
Play aggro early, and use E aggressively, then side step after E to bait her Root. Stay in E range of Neeko because she out ranged you, so if you are in E trande, you can trade with her at all points in the lane.
Your harriers reveal which neeko is real.
Post 6, E her when she ults to bounce of of it.
Like any Mage matchup, if she uses Q on the wave, instantly vault for a trade because that is most of her damage.