Parents Elizabeth Handy and Bilal Walk want to name their 22-month-old daughter ZalyKha Graceful Lorraina Allah.
But state officials at the Department of Public Health said that under Georgia law, the child's last name should either be Handy, Walk or a combination of the two, according to the The Atlanta Journal-Constitution newspaper.
The Georgia branch of the American Civil Liberties Union filed suit against the state on March 23.
"Government has no business telling parents what they can and cannot name their children," said ACLU of Georgia Executive Director, Andrea Young.
The parents said they said gave her the child the name Allah because it was "noble" and it had nothing to do with religion.
The parents said that because of the state's actions, they cannot get a birth certificate for the child and therefore they cannot obtain a social security number either.
They also fear they will have problems with access to health care, schools and travel, the paper reported.
"It is just plainly unfair and a violation of our rights," Walk said.
Young of the ACLU said: "The department's actions interfere with the couple's right to raise their child and are a clear violation of the right to freedom of speech and the right to equal protection under the law.".
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