I was a GA resident all my life. I recently graduated from a GA university and have never been employed. I was offered a job in the State of WA in July. I signed a lease on July 22nd on an apartment, registered my vehicle in WA, obtained a WA drivers licence, etc. I signed my employment contract on July 27th and received my first paycheck in August. I am now being told since I was a GA resident for a portion of the year that I will owe GA income taxes on my earnings, even though I live and work in WA (they do not have state income withholding taxes in WA) Can this be true?

You do not owe GA any income tax, assuming that you had no income prior to moving out of state. But even so - you must still file a GA part-year return. Per the instructions for GA Form IT-511: " Part Year Residents. If you are a legal resident of Georgia for only a portion of the tax year and are required to file a federal tax return, you are required to file a Georgia tax return."

When you complete the part-year return (including Schedule 3) you will indeed report all your income for the year, and calculate a preliminary tax based on that income. But then you multiply the tax figure by a percentage that represents the portion of your full year income that is Georgia source. In your case that percentage is 0%, so your final tax amount is $0. But even though it's $0, you still have to file the return. Details are on page 17 of IT-511 instructions:
https://etax.dor.ga.gov/inctax/2011_forms/TSD_Form_IT511_Instructions_TY2011.pdf