Can't get Arcade's quest
Initiating companion quests in Fallout New Vegas
I cant seem to get Arcade Gannon's quest; can I still get it if I'm a member of the Brotherhood of Steel?
Here is how the quest is initialized:
This quest is initiated by Arcade Gannon, when he realizes that his old friends from the now-defunct Enclave could make a difference in the battle for Hoover Dam.
The quest is triggered by having Arcade as a companion, gaining his trust, and progressing to an appropriate point in the New Vegas storyline. If the player is following the main quest line for an independent New Vegas (doing the Yes Man quests), the quest will be offered even if the player has not gained Arcade's trust.
Here is a detailed link to starting the quest line, this includes how you gain his trust and where in the story line you must have progressed to.
Hope this helps!
Edit: If you have already completed these there is a bug that prevents him from giving you the quest. Here is the work around:
It is possible to not be able to activate the quest for some unknown reason, despite attempting all trigger options, if you're traveling with ED-E in your party get rid of him and the quest should work