Why can't you install the drivers?
Yes. There are a couple possible options. You might be able to use the other Arduino board as a USB to serial adapter board by connecting the boards via their RX and TX pins.
There are some potentially tricky things though. The Nano must be reset at the start of the upload to activate the bootloader. Normally, this is done automatically by the Arduino IDE. You could possibly achieve the same thing by connecting the reset pins of both boards together. Otherwise, you will need to manually reset the Nano at the right time during the upload.
The other possible problem is that the sketch on the adapter board might interfere with the communication.
The other option is to use the spare board as an ISP programmer, using the Arduino IDE's Sketch > Upload Using Programmer feature. You can learn about that here: