Just for future reference, this question would go in the gameplay forums.
But, to save you the trouble of re-posting, the answer to your question is no. There is no quest involving the prisoners in the Vault City detention center.
Those are the prisoners you meant, right? Well, just to be safe, if you're referring to the Vault City Servants, the only quest involving them is to earn freedom for Joshua the servant. Just talk to the distraught woman in the tent directly adjacent to the couryard bar to obtain the quest. (I don't mean to patronize you, I'm just being sure.)