HC-06 Bluetooth Module on Arduino Nano
I agree with TisteAndii's comment. Your voltage divider is in the wrong place. You are dividing down the 3.3V signal from the HC-06 to be a maximum of 2.269 volts. However the Nano requires at least 3.5V to be considered HIGH.

Plus, you are putting 5V from the Nano into the HC-06 which may damage it.