The four are accused of an “abuse of power” while working for a mobile phone company for having passed on private information about subscribers and their call history, Azeri prosecutors said.
During a search of the home of one of the accused, investigators discovered “religious literature, disks, brochures containing speeches by Fethullah Gulen and other documents,” they said in a statement.
A close ally of Ankara, Baku earlier this week opened a criminal investigation into supporters of Gulen, who is accused by Turkey of ordering the July 15 coup in a bid to remove President Recep Tayyip Erdogan from power.
Gulen has consistently denied any involvement in the failed putsch.
Azerbaijan last month shut down a private television channel over plans to broadcast an interview with Gulen, “in order to avoid provocations aimed at damaging the strategic partnership between Turkey and Azerbaijan”.
Gulen’s Hizmet movement has affiliated schools around the world, including in ex-Soviet Kazakhstan, normally funded by wealthy Turkish businessmen.