Short version... No.
Longer version...
Technically GSM supported fax but through a special profile. You have to make a circuit switched connection - which is a 9600 baud connection over the cellular network. Most modern GSM handsets do not have this feature enabled because we all use LTE, UMTS, HSPA, etc. Back in the day (mid-90s), I used to have to do circuit switched connections over cellular to dial into various PBXes and work on them. Connecting to the GSM handset for that purpose requires some specialized hardware - that is proprietary to each different phone and not made for modern handsets. The easiest way to connect a fax machine would be through a tellular unit - which is basically a GSM phone in a box with an RJ-11 jack on it... that's if you could find a provider that would give you a circuit switched connection. The new HSPA and LTE base stations don't support it. So you'd have to be on an OLD GSM network for it to work.
If you go look at the specs for the Verizon Wireless Home Phone Connect and similar units from AT&T, you will see that those boxes (which have RJ-11 jacks) specifically do NOT support fax machines.
The reason that you can't connect a fax to any new cell phone and have it work is a matter of compression. The codecs used on your phone are heavily optimized for voice. They aren't made for dealing with the blips and burps made by your fax machine or a dial-up modem. That's the same reason that you can't send a fax over VoIP using a lossy codec like G.729.
So effectively, the answer is you can't do what you think you want to do. You're either going to have to suck it up and pay more to use a traditional phone line and fax, or you're going to have to suck up your (faux) security concerns and get over the idea of using a 3rd party fax service.
}Davoice