Do you think that Hermione would become jealous of a girl if that girl was Harry's girlfriend?
No, I really don't think so. Even if Hermione is closer to Harry than Ron in your story, I think that they're just friends. Harry went out with Cho Chang and Ginny Weasley. Hermione didn't really show signs of jealousy towards either girl. She seems to be fairly neutral when it comes to Harry's... relationships, I guess you could say. We haven't seen any signs of her having difficulty putting up with them.
Would she try to turn Ron against this girl?
I can't really see her doing that. From what I've seen, there are people she dislikes, but she doesn't seem like the person who tries to persuade others to hate someone else and go against them, you know what I mean? If she wanted to turn Ron away from the girl, it wouldn't be out of jealousy, unless this girl was unfairly beating her in every class or something (think Hermione's reaction to Harry's success in 6th year Potions).