On all the boards I've worked with, the Serial object is "bound" to a certain pair of rx and tx pins. Some boards do indeed have multiple serial ports, but those boards have multiple objects(Serial1, Serial2, etc), each bound to a specific set of pins.
Serial communication is so popular that it's handled by hardware in the chip, and in general, on low end microcontroller, each hardware module for serial stuff is bound to a specific set of pins.
Various serial port info, most of which you probably already know, but included anyway for completeness.
Some chips let you actually select what pins connect to the UART, but the chip in an arduino uno is not one of those to my knowledge, and if it is, the arduino library doesn't have a way to manually change the setting and you would need to do some low level register writing type of code. Pretty sure the atmega328 just plain doesn't have that feature though.
The arduino uno has only one serial port though.
You can use software serial to create fake serial ports, manging dataflow with clever code instead of special hardware. You could hook the module up to the virtual serial port and use the real one for talking to the PC. Soft serial is included in the library. For this I think you do need to define what pins to attach the virtual software port to, because you actually do have a choice.
The name COM4 is purely on the computer side. The arduino board contains a USB to serial chip that interfaces between the computer and the arduino. When you plug the USB cable in, Windows creates a virtual serial port, which appears to software like as if it were an old style rs232 serial port. It assigns the virtual port a name, something like COM1, COM4, COM752, etc.
The Leonardo has the usb to serial built into the chip, and it has a normal serial port as well, giving two ports, and the mega has a ton of serial ports.
Alternatively, if you aren't going to send anything to the Bluetooth, and you aren't going to receive anything from the computer, you can use the RX pin for the Bluetooth module, and use the TX pin for the computer. You would be splitting the two hacks of one port into two one way ports.