State Taxes

Hi! 

My wife works from home for a non-profit organization based out of Washington (no state taxes), but she lived in Tennessee (no state taxes) for four months and then moved to Georgia (state taxes) for the rest of the year Her company did not take out any state taxes for her. In this situation, do we have to fill out an additional form, calculate the taxes owed based on how much she made while in Georgia, or something else? Her company also stated that she should get a credit from Washington, but we have no idea what that means. 

Thanks!