As long as no one else claims the child as a dependent on another federal return, you could claim that child on your state return. To do that, you would have to either prepare a manual state return or use a TurboTax desktop (CD/Download) program and prepare two returns, one with the child as a dependent on both federal-state return and the other without the child.
The only reason to claim a child on your state return and not on your federal return is if someone else is claiming them for federal purposes. If the child qualifies as someone else's dependent on their federal return, then only they would be able to claim a dependent exemption on a state return.
