Arcade's quest is unusual in that it won't start until late in the main quest. Also, you have to take him to certain places first which trigger special conversations with him. These conversations reward 'friendship points' and you need two to for him to start the quest. Even then, you must wait until later in the game for him to mention it.
As for scoring friendship points, the easiest is simply entering Caesar's fort, tell him you just want to listen to Caesar and then leave, and that wil earn you three points.
Basicly...
1. Enter the Fort with Arcade.
2. Arcade will complain and you'll have to tell him you won't be there long.
3. Wait until later in the game when he'll eventually tell you his big secret and start the quest.