I don't see them being close at all (and it's reflected in my story).
There's nothing in the lore to suggest they have a close relationship. If anything, Lux's judgement shows that Garen is rather indifferent (if that is actually him and not an illusion created by a summoner), and Lux seems to have no inclination that way either. She doesn't even greet him upon seeing him for the first time in years.
In my personal opinion, Lux might actually resent him for disappearing and not contacting her even after she 'joined' the military. Wouldn't you be a little upset if that happened?
On another note, Lux is rather selfish, and is focused on her own goals rather than Demacia's, unlike her brother. Garen lives for Demacia, and will die for Demacia. As much as Lux loves her home, she hasn't decided what's really important to her yet.
So please don't fall for the whole 'more than siblings' thing. It has little to no base. :/