HC-05 Bluetooth Module - INQ-Command doesn't work
I did a factory reset of the device. That resolved the issue

AT+ORGL
AT+RESET


Another thing that is very important: The KEY-Pin (EV-Pin on some boards) must be HIGH (3Volts), during the INQ-command. If this PIN is not High, the inq-command will not work.

If you put 3V to this PIN and release it, you indeed are in AT-Command mode. But not all AT-commands work (e.g. the said at+inq command). If you want to enable all AT-commands the PIN must remain high.