I immigrated into USA in February 2004, worked in Georgia for about 5 months, then moved to Texas and have been a residence of Texas ever since. I would like to know if I need to file a State Tax Return for the 5 months I worked in Georgia. Thank you.

Yes, you file tax return for Georgia as part-year resident for the period you were in Georgia. Any income earned while is Georgia must be reported on the Georgia tax return. You file only if the income exceeds the filing requirement.
Read: Your U.S. Tax Return: Working or Living in Two or More states

Also, if you had refund due from Georgia, you have FORFEITED that refund, because, like the IRS, Georgia has a three-year statute of limitations for claiming tax refunds.