No and maybe. Child support payments are neither deductible by the payer nor taxable income to the recipient. The payer of child support may be able to claim the child as a dependent:

• If the payer is the child’s custodial parent for federal income tax purposes, the payer is generally the parent entitled to claim the child as a dependent under the rules for a qualifying child if the other tests for claiming the child are met. The custodial parent is the parent with whom the child lived for the greater number of nights during the year. The other parent is the noncustodial parent.
• If the payer is the noncustodial parent, then the payer may only claim the child as a dependent if the special rule for a child of divorced or separated parents (or parents who live apart) applies. That rule requires, in part, that the custodial parent sign and provide the noncustodial parent a Form 8332, Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent or a substantially similar statement. The noncustodial parent must, then, attach a copy of that release to his or her return in order to claim the child as a dependent.