Can I put a fan control switch on a window air conditioner to quiet it down
Assuming the motor for the AC fan is of the proper type, "shaded pole" or "PSC", then yes, if you opened up the unit and tapped-in to the fan control circuit itself.  This device would not work if you tried to power the entire AC unit from it.

I suspect, however, that the motor is neither of the above types.  Further, you are looking at significant safety risks by modifying a self-contained AC unit in this way.  It would certainly void any manufacturer warranty and may also create a safety hazard.

Most window AC units I've seen have a built in speed control.  Is there some reason why you can't just use what is designed into the unit?