I had the same problem in getting the call for this achievement on both 360 and on the xbox one.
I bought the taxi company and went many hours (real time) playing around as Franklin without getting any call. Now this may have just been a coincidence, but I found if you buy the towing company, I got a call within 30 minutes real time in game for a private fare.
In order to purchase the towing company, you have to complete all of Tonya's towing missions as Franklin. You also need to call her at some point to get the next mission as it won't automatically give you all the missions consecutively.
The first time I was supposed to be a getaway driver for a guy robbing a store, and he told me to wait in an alley with the car, but he got shot by police before even reaching my car! So it was a failed fare. The second customer I got was a passenger wanting to go confront the Lost about his niece. If you get this mission, MAKE SURE YOU GET OUT OUT THE CAR WHEN HE DOES! As soon as the Lost draw their guns, kill them before they kill your passenger! After that he thanks you and advises you leave before the cops show up. Mission complete and achievement unlocked.
These are the only two missions I tried, so I am not sure if you will get something else, but hopefully this will be helpful to you.
Best of luck!