No, you can't - only the parent claiming the child as a dependent can actually claim EIC. You and your girlfriend are allowed to agree who will claim the child.
Note: The rules are slightly different for parents who are not sharing the same address (it includes separated and divorced parents). The custodial parent (parent with whom child lives the longest) can claim EIC, dependent care credit and Head of Household status) even if he/she allows the noncustodial parent to claim the child as a dependent.
