There are two ways to "win" that competition. The first you have to brace yourself and suffer through what happens with Quinn, which results in you winning the competition.
The second option is even more rewarding (if you don't like Quinn) as it frazzles her and throws her off, but it's a bit costly because you have to lose the actual competition itself. But if you want to see Quinn lose her cool (and in her mind...lose the encounter), DON'T win the competition. It's pretty awesome to watch at least once in a play through. I found out-playing Quinn to be so rewarding that I kept the Loss in my Main Save.
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