I don't think there is an established E.S. metric for counts. Conventional ES has to do with a standardized mean difference for 2 groups for a continuous outcome and you don't have a continuous outcome. One possibility that is relevant because count distributions are characterized by probabilities is to look at group differences in probabilities for e.g. count = 0. See Chapter 6 of our RMA book.
You can look at count mean differences but that is more abstract.