Get your taxes done using TurboTax

Yes, both options are acceptable, you could claim 100% or you could claim 50% and your ex-wife claim 50%.  You can split the reporting of the State Refund from the 1099-G between the two of you any way you and your wife agree upon.  

In most cases, Ex-Spouses each report 1/2 of the refund to keep things simple.  However, feel free to agree on splitting it in whatever way seems fair to both of you.