The state Department of Public Health has refused to issue the 22-month-old with a birth certificate.
Elizabeth Handy and Bilal Walk say it is unacceptable that their child has officially been left nameless.
But state officials say the child's surname - ZalyKha Graceful Lorraina Allah - should either be Handy, Walk or a combination of the two.
Allah is the Arabic word for God.
The American Civil Liberties Union (ACLU) of Georgia has filed a lawsuit in Fulton County Superior Court on the family's behalf.