Justice of the Peace Marriage in Church?

Generally speaking, the answer would be no. Catholic churches are for Catholic liturgies. Churches are sacred space, and secular events within them should be greatly limited (Code of Canon Law, can. 1210).

The only situation I could possibly see happening would be in a part of the world where there are very few priests and deacons. In such a case, the local bishop could delegate the authority to witness the exchange of vows to a competent layperson (can. 1112). However, in those circumstances, the fact that the individual is a justice of the peace would be merely coincidental.