No, paying child support doesn't mean that you can claim your child as a dependent. If your child spent half the year -- 183 nights or more -- with the other parent, then they are the custodial parent and they are generally entitled to claim the child as a dependent.
However, please see a previous AnswerXchange response, below, for additional information that might help you.
https://ttlc.intuit.com/replies/5679672
