We propose a method of spatial prediction using count data that can be reasonably modeled assuming the Conway-Maxwell Poisson distribution (COM-Poisson). The COM-Poisson model is a two parameter generalization of the Poisson distribution that allows for the flexibility needed to model count data that are either over or under-dispersed. The computationally limiting factor of the COM-Poisson distribution is that the likelihood function contains multiple intractable normalizing constants and is not always feasible when using Markov Chain Monte Carlo (MCMC) techniques. Thus, we develop a prior distribution of the parameters associated with the COM-Poisson that avoids the intractable normalizing constant. Also, allowing for spatial random effects induces additional variability that makes it unclear if a spatially correlated Conway-Maxwell Poisson random variable is over or under-dispersed. We propose a computationally efficient hierarchical Bayesian model that addresses these issues. In particular, in our model, the parameters associated with the COM-Poisson do not include spatial random effects (leading to additional variability that changes the dispersion properties of the data), and are then spatially smoothed in subsequent levels of the Bayesian hierarchical model. Furthermore, the spatially smoothed parameters have a simple regression interpretation that facilitates computation. We demonstrate the applicability of our approach using simulated examples, and a motivating application using 2016 US presidential election voting data in the state of Florida obtained from the Florida Division of Elections.