Felix knew that his friendship with the other second in command was a shock to Henry, but there was a very real reason he decided to make an effort to get on the other knight's good side.
Alix was sent to the north on behalf of Commander Lothian to make sure that Henry was taken care of and that the knights he came across weren't disrespectful in any way. He was also there to make sure that Henry paid attention to his own needs while throwing himself into the needs of the Chemoian people as well.
Becoming friends with Alix was the only natural thing to do because he was like a roadmap to the knight Felix was supposed to call commander. Whenever he tried to get under the surface of Henry, it was clear something was holding the knight back. He wondered if it was his life as a lowborn that had caused him such mistrust - perhaps Chemoian knights in the war had mistreated him and Felix wasn't aware of that since he had other things to worry about.