Ok well I use to have Arcade Gannon as a follower until I told him to go to the Lucky 38 and stay there and I replaced him with boone. Well I found out the reward for the "For Auld Lang Syne" quest is the Remnants Power armor, power armor training, and possibly the gannon family tesla armor. Well I already have the Remnants power armor but I just want to do the quest because I have found out how much you get. Well I went back to the lucky 38 and to talk to arcade gannon because thats who you need to talk to start the quest and there is no dialogue option to start the quest. I think I have earned enough trust from him to do this and I am unsure what to do. Should I tell him to go back to the old Mormon Fort because i thought it may be because he is in the lucky 38 and an ex-follower. I am not sure anyone else know how?

You have to reach the point in the main quest where you get the message that there is no going back. Just say you aren't ready, and talk to him.

I think it is actually available at any point after dealing with the brotherhood of steel, you wont be able to choose dialogue, he will randomly run up to you while he following you around and then begin to talk about it

The Wiki says otherwise. And from my experience, the Wiki is right in this case. I've only gotten the quest from Arcade when I reach that point in the main quest.

That is not what the wiki says. For The Republic Part 2 (or the specific example of between the Omertas quest and just hearing about the Brotherhood of Steel) are a quest or two prior to the point of no return. It says committed but that's misleading, all its details are prior to the point of no return.