Q could have got what he needed from Kathryn's dandruff Brush.
"WE'VE ALL BEEN THE SCARECROW!"
Q knew his child, and knew who the mother of his child was, but causality insisted that he had a role to play, and that he couldn't get it on with Q until after Janeway had rejected him, and frankly he couldn't risk ever that she might not reject him if there's a little free will wiggle room in proximity to a Q there is no where else in the universe.
It's called Omniscience.
So not only was he after harvesting DNA more so than making love, Janeway was a fallgirl.
Romeo was after the wrong girl for the first couple acts too.