We have a solution to fix staff's unable to faxing on Quantum eCW per eCW Tech Support Team. The issue is nothing to do with permissions but rather has to do with missing association with "Default Appointment Facility".
If left blank, will cause the issue unable to fax. It's their programming so we just have to go with it.

To access to this setting on indvidiual level, not something we can do as admin but it's personal settings:

On your Quantum eCW:

File  > Settings  > My Settings  (the bottom panel will open up) then click on the "My Providers" tab.
My Providers > CLICK ON   "..." next to "Default Appointment Facility" and select a facility to server as a default.
Select one Quantum Facility, preferrably your default location, It will ask you if want to change:  Click "YES" and restart eCW.

If YOURS is missing or blank like one staff's was, this is the reason why unable to fax out for that patients.   Just how eCW works. ¯\_(ツ)_/¯