It has already been reported that we received a tax refund from Virginia last year. Do I still enter the 1099-G from Virginia?

That 1099-G is probably for the refund you received in 2016 (and should be the same amount). So as long as you have entered the amount of the refund your received in 2016, you do not have to enter that 1099-G refund information again. If you do, it could potentially be "double-taxed".