Tribbles will breed in your character's device slots, as well as in inventory and the bank, but you need to be active in person (i.e. on a planet or station) when the feeding timer ticks if you want device tribbles to breed.
Since active BOffs will immediately eat any food you give them, using them to breed tribbles is not possible. (Either they'll be inactive and thus ignore the feeding tick, or will have consumed the food.)
(I haven't investigated how shipboard activity affects tribble breeding.)
Tribbles breed "on the hour" for your character. To find out when that is, type /played into the chat window. So if you have been on duty for x days, y hours, and 45 minutes, the tribble will breed in 15 minutes.