You can probably try this out for yourself, with the equipment you have kicking around...
It's a kludge, and perhaps not worth the effort, but if you are really determined, here goes. There's a good chance you already have a 'real' modem on your PC, with a standard RJ11 telephone port, normally used to 'dial out' on a telephone line. Connect this modem signal to a spare FX0 port on your PBX. Configure the DID to route to this FX0 port, to the modem. The modem also would need to setup properly, and wouldn't be trivial.
Now here's the kicker. A lot of people have been fairly successful in getting FAX to work over VoIP, some not so. FAX signals are analogue "tones" that must be faithfully reproduced over a voice channel, in this case VoIP. Consider that the transmission data rate for typical FAX connection is limited to 19.2 kilobits/sec, and if the voice circuit is 'noisy', the rate can be negotiated even lower.
You undoubtedly recall the sounds that a dial-up modem makes when initially connecting to an Internet Service Provider? These squeals and tones that are heard are the two modems, end-to-end, negotiating the best data rate for the current line conditions. Sometimes you get lucky with a good 'clean' line, and see blazing speeds like 56 kbs, or maybe as low as 28 kbs with a noisier line. Can you imagine what Internet performance would be like if the BEST you could do was only 19.2 kbs? I'd be ripping out my hair!
Of course, YMMV, so if you still feel inclined, go ahead. I just wanted to give you a preview of "the prize" you are shooting for.
Cheers