Officers said the men, aged 20 and 23, were reported to police by residents in the Rukoh area of the provincial capital, Banda Aceh, on March 29.
Marzuki, the chief investigator, said the two men had "confessed" to being a gay couple and that this was supported by video footage taken by a neighbour.
Aceh is the only province in Muslim-majority Indonesia to practice Shariah law, after a concession made by the national government to end a years-long war with separatists.
A Shariah code implemented two years ago allows up to 100 lashes for "morality offences" including gay sex.
Caning is also a punishment for adultery, gambling, drinking alcohol, women who wear tight clothes and men who skip Friday prayers.
Marzuki said residents in Rukoh were suspicious of the two men because they often seemed to be intimate, and had set out to catch them having sex.
He said: "Based on our investigation, testimony of witnesses and evidence, we can prove that they violated Islamic Shariah law and we can take them to court."
Homosexuality is not illegal in Indonesia, but a judicial review being considered by the Constitutional Court is seeking to criminalise sex outside marriage and sex between people of the same gender.