Janeway had a similar question:
JANEWAY: I don't see how a baby is going to end a war being fought by a race of omnipotent beings.
Q: It's simple. Mating will create a new breed of Q, which will combine my omnipotence and infinite intellect with the best that humanity has to offer.
JANEWAY: You believe human DNA is going to restore peace?
Q: Precisely. What the Continuum needs right now is an infusion of fresh blood, a new sensibility, a new leader, a new messiah. Think of it, Kathy. Our child will be like a precious stone tossed into the cosmic lake, sending endless ripples of human conscience and compassion to wash up on every distant shore of the universe. What greater contribution could a being of your limited power ever hope to make? What is more important to humanity than peace? I'm offering you the opportunity to be the mother of peace.
As it was mentioned above, Amanda Rogers may have fit the bill so to say in this regard... but, is it possible that she was one of the first casualties of the Q Civil War and also shunned by the Continuum given her origins?
It took quite a bit of convincing for Q not to kill her (obviously), and Q even suggested at some point she's neither Q or a Human... and was 'disgusted' by the notion she could be something in between.
I don't understand this fetish for some writers to have their caracters being one thing or the other. Why can't a person be something in between?
Especially when you consider the fact that there's Torres who's a Klingon/Human hybrid... or Worf who was born a Klingon but grew up in the UFP and alternated for a while between the two, but in DS9 he acknowledged he lived in UFP for too long so he considered his brother request as 'murder'.
Rodenberry (or should I say, the underlying issue that UFP is an open and accepting society) wouldn't have an issue with individuals who cannot define themselves clearly and would be perfectly fine with people being 'something in between'.
So, why couldn't the Q get over this idea?
If I remember something from when Quinn comitted suicide ('Death Wish')... he often spoke about 'purity of the Q' and how he even celebrated what they were, and then started questioning the Continuum in ever increasing quantities (for which he was imprisoned).
I guess the Continuum lost a few things on their evolutionary path - maybe its some of those things that Q interested in UFP and Humanity... and specifically Picard (though, some would say Q fell in love with him).