Can a Solid-State Relay Control AC Powered Fan Speed Via PWM
If you pulse the supply on and off to an AC motor, then it will run slower.

However, as happens so often in technology, the problem isn't getting what you want to happen to happen, it's getting it to happen without unwanted side-effects.

There are only 100 or 120 half-cycles (depending on locality) of AC supply. If you use a zero-cross SSR and start dropping out every 3rd cycle, then as well as turning slower, the fan will also likely to emit a loud humming noise, louder than it normally does. This may or may not be acceptable to the listeners in the room. Only odd cycle-drop sequences are permissible, if you go for even sequences, you will supply net DC to the motor, which will saturate it and quickly burn it or blow a fuse.

If you use phase-shift control, then the hum it emits will acquire lots of high harmonics, much more audible and annoying than the low hum.

If you can actually chop the mains PWM fashion, then it will emit an audible whine at PWM frequency, unless you chop at a high enough frequency. Depending on the motor quality, it may get very hot with PWM, especially high frequency, remember these are built down to a price not anticipating use like this.