Can you play a LAN game between 2 different computers if your router is not hooked up to the Internet this game does not require the Internet when playing between the two computers

LAN games with a router Yes-

A router which has more than one ethernet socket (usually marked "LAN"), where your computers plug in has a built-in "hub" or "switch" function which allows all the devices plugged in to communicate between them.

Thus two computers plugged into the same router will be able to communicate with each other even if the Internet connection (which comes through the actual router and is also connected to the internal hub) is not plugged in or active.

This will allow "file sharing" between the two computers with Windows, and also will allow the computers to play LAN games.