Do other Bluetooth devices show up on your phone, for example wireless keyboard or earphones? Can other computer or phones see the Bluetooth signal of the phone?
Before you set the phone back to factory settings, try to reset the network settings in Settings/General/Reset.
If resetting the network settings does not work, and to make sure that this is not software related, set it back to factory settings, without using any backup data afterwards. Set up the rest of the personal settings manually and test the Bluetooth connection.
If the Bluetooth works, restore from the latest backup. If the problem comes back, the backup data contained corrupt data, causing your issues.
If it still does not work, this is a hardware issue and the phone has to be serviced by Apple or an Apple Authorized Service Provider:
Use iTunes to restore your iOS device to factory settings - Apple Support
Find an Apple Authorized Service Provider
iPhone - Contact Support - Apple Support