Investigating relationships between response variables and covariates in environmental science, geoscience, and public health is an important endeavor. Based on a Bayesian mixture of finite mixtures model, we present a novel spatially clustered coefficients regression model for count value data. The proposed method detects the spatial homogeneity of the Poisson regression coefficients. A Markov random field constrained mixture of finite mixtures prior provides a regularized estimator of the number of clusters of regression coefficients with geographical neighborhood information. An efficient Markov chain Monte Carlo algorithm is developed using multivariate log gamma distribution as a base distribution. Simulation studies are carried out to examine the empirical performance of the proposed method. Finally, we analyze Georgia's premature death data as an illustration of the effectiveness of our approach.