Guess i maybe don't understand the question at all. Quite confused. FIOS is Fiber Optic service not wireless. So not clear what you mean by "since the fioz modem was plugged into the external antenna." The Fios modem connects to a Optical Network Terminal (ONT) box usually outside your house at the service panels which terminates the fiber optic link and converts it to copper for distribution in your house. It doesn't connect to an antenna.
4G LTE is wireless and operates at the 700MHz band (LTE Band 13). Forward Link = 746 - 756 MHz, Reverse Link = 777 - 787MHz. Because this is somewhat of a low frequency (compared to for example WiMax at 2.5-2.7GHz) the coaxial cable quality is not too critical because the signal attenuation is proportional to frequency (can get away with LMR200 for example). Needs to be 50Ohms though so the 75Ohms CATV RG6 cable is no good.
If you are referring to satellite Internet then even this would not work because the satellite dish operates at a totally different and much higher frequency. For example Ku band (12 to 18 GHz), K band (18 to 26.5 GHz), ka band (26.5 to 40 GHz) etc. The IFL coax cable (Interfacility link) for satellite is typically 75 Ohms.
So i guess to answer your question: (1) If you have a 50 Ohm coaxial cable, (2) with the right connectors, (3) and connected to an antenna operating at 746-787 MHz, then yes you can re-use such a setup.