What does "on the nose" mean in the context of horse racing bets?
The character is probably confusing "on the nose" with "by a nose", which means to win by a slim margin. But no one bets on that, they just bet on the position that the horse comes in (first, second, third).
"on the nose" normally means "precisely" or "exactly", so I suspect she's using this to mean that the horse should be specifically the winner, not one of the other positions you can bet on. I can't think of any other possible interpretation.