You'll pay the tax to Jackson County for 2009. The tax is assessed by the county where the vehicle was garaged on Jan 1st.
Stop by the tax collectors office in Miller County and give them your details. They can probably give you a 2010 assessment form and complete it on-the-spot. If the 2010 forms are not ready yet, they'll add you to the mailing list and send you the form shortly after the new year. Be sure to return it by the deadline (Mar 15?) to avoid penalties for late assessment.
If you get an assessment form from Jackson County, send it back uncompleted with a note that you have moved to Miller County. This isn't strictly necessary but will help them keep their records up-to-date.
