Perhaps, yes, it might fry the USB-serial chip on the Nano. Or perhaps that chip will shut down to protect itself and the Bluetooth module will stop working, I'm not sure.
Some hc-06 modules have a built-in 3.3V regulator and so can be run from the same 5V supply as the other components in your circuit. Does your hc-06 module have a regulator?
For example this one:
This module has a built-in 3.3V voltage regulator