QT has native bluetooth support, but can it be used to communicate with the Lego NXT robot?
Sure it can. The NXT provides a bluetooth serial port, so you don't actually have to do anything with bluetooth in Qt. All of the Bluetooth stuff will be handled by your OS.
Instead, to communicate to the NXT from Qt, you will need to open the serial port for reading and writing, then send commands to the NXT over this connection. You can find out more about the available commands using the NXT Hardware Developer Kit.