I would say this took place during Stewie's fantasy when he "supposedly" killed Lois and Quahog was canvassing the neighborhood for him. I mean, I can see how Joe or any Rhode Island cop can mistake the Brain for Stewie.
They both have a small stature, are condescending geniuses with large craniums, and they both have similar ambitions: Although Brain does attempt to fulfill his, even though his plans get foiled every night. Stewie diverts his attention between his attempts on his mother's life and world domination.
Update: I don't know if you'd like this but I added Roger's pic to the wall, in his Tony Montana look from the episode, 'Dope & Faith'.