A reason for saying that “God only knows” is that it would be impious or irreverent to suggest that He (who knows everything) does not know.
For that reason, you get expressions like this one on the Tomb of the Unknown Soldierexpressions like this one on the Tomb of the Unknown Soldier.
A corresponding secular phrase could be simply, "I don't know", "No-one knows", "It's unknown", or "It's a mystery".
