Fortunately for you, Wyoming has no personal income tax system. Therefore you have no income tax return to file in Wyoming (ever, unless you own a business), so you can remove that from you list of concerns right away.
Now then, is your 1099-G related to an Oregon income tax refund that you received, based on filing last year's Oregon state tax return? If so, then you only need to enter it into the TurboTax program if you also itemized your tax deductions in that same year. If you instead took the standard deduction (as more than 70% of taxpayers do), then you can safely ignore the Oregon 1099-G.
On the other hand, if the Oregon 1099-G is instead for the receipt of Oregon unemployment benefits, then you do need to report that on both a federal tax return (Form 1040, 1040A, or 1040EZ) and on an Oregon state tax return.
Which kind of Oregon tax return you'll need (if applicable) will depend on your residency status. If you lived outside of Oregon from January 1, 2016 to December 31, 2016, then you'll file Form 40N as a nonresident. If you lived in Oregon for part of 2016 before moving to Wyoming, then you'll need the Oregon Form 40P.
The TurboTax sofware will take care of all of these mechanical details, as long as you answer all of the program questions accurately.
And, as we discussed above, you should have nothing to file in Wyoming, as it pertains to your Oregon 1099-G, or not.
Thanks for asking this important question.
