No, do not report 1099-G income that was fraudulently obtained in your name. If you report it on your return, it may result in additional tax.
Be prepared, however, to get an automated notice from the IRS later in the year. If the 1099-G income would result in an increase in your taxes, the IRS will send you a CP2000 Notice of Proposed Change in your taxes. The instructions allow you to agree with the change (and pay the tax) or disagree with the change, by giving your reasons why you don't owe the additional tax.
See the following article for an excellent discussion on how this notice system works: How To Read And Respond To Your Notice From The IRS