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.
The girl's father told the Atlanta Journal-Constitution they had called her Allah because it is "noble".
"It is just plainly unfair and a violation of our rights," Mr Walk said of the state's refusal to acknowledge the name.
However, lawyers for the Department of Public Health said Georgia code "requires that a baby's surname be either that of the father or the mother for purposes of the initial birth record".
In a letter to the family, state officials wrote that ZalyKha's surname can be changed through a petition to superior court, but only after the birth record is recognised.
According to the lawsuit, the unmarried couple already have a young son called Masterful Mosirah Aly Allah.
The ACLU says that without a birth certificate the parents cannot obtain a Social Security number for their daughter.
They fear the girl's identity and rights as a US citizen will be questioned.
The ACLU said the state's refusal to grant the family's wishes was an unconstitutional example of government overreach.
"The parents get to decide the name of the child," said Michael Baumrind, a lawyer for the family. "Not the state. It is an easy case."