In Washington, the state's UIFSA (Uniform Interstate Family Support Act) statute applies to situations where there are competing child support orders issued by two different states. You say that there was a child support motion filed in 2011 in WA, but it is unclear whether the court in WA actually issued an order of child support. The first step will be to figure out if there are two court orders (one from a court in Washington and one from a court in Georgia) addressing child support or...
It will depend on the specific facts of your case; paternity matters can be complex. However, challenging paternity is generally started by asking the court (by filing a legal action) to adjudicate parentage pursuant to Washington's Uniform Parentage Act.