And, if so how much? Will they go up if rates have already gone up due to an earlier accident?
I doubt it, so long as you didn’t require medical help. My wife was in an accident that was partially her fault (she wasn’t ticketed, though), and our car insurance rates didn’t increase due to the accident itself. However, as per the insurance rep — they did increase because of the large hospital bill. We ended up switching providers because that’s a bunk policy.
You could always call your insurance company — most of them will be able to tell you one way or another whether certain things will increase your insurance rate.