If you are physically working in Montana then you will have to file a non-resident Montana state income tax return to report the income that you earned in Montana.
Washington does not have a personal income tax so no Washington state tax return is filed.
Since you do not live or work in Montana, you do not have to file a Montana tax return. But giving your employer a Montana address will raise questions. The IRS shares information with the states. If your Montana address is on your W-2, or if you use the Montana address on your federal tax return, it will look like you are a resident of Montana. You will probably get notices from the Montana Department of Revenue saying that you have to file a Montana tax return and pay Montana income tax. You will have to be able to prove to them that you do not live or work in Montana. It would save you a lot of aggravation if you use your Washington residence as your mailing address.
If your employer has any presence in Montana, and you give them a Montana address, they might be required to withhold Montana tax from your pay. Then you would have to file a Montana non-resident tax return to get a refund of the tax that was withheld. And, again, you would have to prove to the Montana DOR that you did not live or work in Montana.
Why do you want to give your employer a mailing address that is not where you live?