The cable does not care or have 'compatible' questions, its only the connector that matters.
The TV coax will have an F connector, so you just need to adapt it.... Since you are going to run an LNA (yes, you are _going_ to run one, no question), then really its a matter of going from your SO239 to the LNA and from the LNA to the TV coax.
Here is the LNA:
https://www.amazon.com/Nooelec-SAWbird-NOAA-Ultra-Low-Applications/dp/B07TWPR871/

Then from the SO239 female (Assuming your antenna has a male plug - you never mentioned) you will need this:
https://www.amazon.com/DHT-Electronics-coaxial-adapter-female/dp/B00CVQOOAI/

Then from the LNA to the TV coax:
https://www.amazon.com/DHT-Electronics-coaxial-adapter-female/dp/B00CVTRTUC/