You only have to report your income on the Oregon state income tax return for the time you were actually working in Oregon.
In the My Info section of the TurboTax online program you are asked if you made money in any other states. You would answer Yes and select Oregon as the other state.
You will complete a nonresident Oregon state income tax return reporting the income earned while working in Oregon and all the Oregon state taxes withheld from your wages. If you overpaid the state taxes based on your income you will get a state tax refund.