If by DEQ you mean the state emissions test, you can register your car without it. We've done it a couple times. Go to the Washington DOL web site and look for the emissions waiver form. It must be notarized which can be done at the consulate in Nuevo for a price ($30 last time I did it but I think it's gone up). Another alternative, if you return to Washington without the car, is to go to a local license agency where you can sign a notarized declaration that the car is out of state and then you can renew your license.
"Live in the sunshine, swim the sea, drink the wild air…” Ralph Waldo Emerson