In this paper we give an algorithm for computing the conjugacy growth series for a right-angled Artin group, based on a natural language of minimal length conjugacy representatives. In addition, we provide a further language of unique conjugacy geodesic representatives of the conjugacy classes for a graph product of groups. The conjugacy representatives and growth series here provide an alternate viewpoint, and are more amenable to computational experiments compared to those in our previous paper. Examples of applications of this algorithm for right-angled Artin groups are provided, as well as computations of conjugacy geodesic growth growth series with respect to the standard generating sets.