I want to make conversation with a girl my girlfriend hates
Why do you want to make conversation with her?
To upset and disturb my girlfriend.
If you're doing this to get back at your partner then it is a bad idea. She will be upset to see it happen, and especially angry to learn that you did it on purpose.
Who cares, who said she would ever discover that I was behind it. I'm going to do it anyway.
One way or another your girlfriend is probably going to find out that you had this conversation, and a minute ago you seemed to like the idea of upsetting her. However it may turn out, I have to warn that this is a bad idea, and it will just cause unnecessary hardship. If you are upset about something, you need to let your girlfriend know so that you can both fix it.