This is super clunky and going to be very slow.
http://h10032.www1.hp.com/ctg/Manual/c01420218
Says the max speed is 14400 baud (bits per second)= 1800 bytes per second.
Computers used to have FAX/modems
https://www.amazon.com/Robotics-USR5637-Controller-Dial-Up-External/dp/B0013FDLM0
If you found one of those you might be able to connect the computers FAX modem to the FAX machine.
However, there is a chance it won't work because it can't detect the dial tone.
You might have to use some kind of Y phone cable to introduce the dial tone.
Also you will have to find fax software to convert your document into the FAX format, and that will also introduce a bunch of quality loss intentionally to make the file size smaller.
Even if all of this does eventually work, print quality is going to greatly suffer as most modern printers use 300+ dpi, and the fax is likely to use 75dpi.
Apparently your device supports:
Super Fine: 392 x 203
It will however, be incredible slow.
My best guess as someone who used modems regularly in the 1980's and 1990's that once you see how slow the output is you will just end up buying a new printer.