The exhaust fan is permanently powered. Its speed is controlled via the PWM input signal, and although that does indeed go to 100% (DC) while operating (i.e. it is not variably controlled), the control circuit almost certainly can not source enough current to drive a relay or similar. So you would need a circuit capable of converting the PWM signal to a suitable level for driving a relay. Technically, the way it is being used, that could be as simple as a transistor, as long as they don’t change the way it’s being driven (i.e. use the fan’s PWM speed control as intended by the manufacturer),
TL;DR - this can be done as things are today by utilizing the PWM speed control signal (yellow) with a transistor/MOSFET circuit.
I’m an EE. I prefer to use a WeMo smart switch, and just tell Alexa to turn on my Glowforge fan.