The standard Poisson Auto-Regression framework considers static coefficients and does not incorporate any spatio-temporal dependence on the parameters governing the process dynamic. However, unobserved space-time variability is a very relevant component when dealing with observations organised in space and time. We consider a more flexible specification that can adjust for local deviations from the general pattern while borrowing information from adjacent areas. The model, specified in a Bayesian framework, might suffer from computational bottlenecks that can make its estimation unfeasible. Therefore, we implement it in STAN to jointly update all the parameters and improve mixing, while adopting a novel sparse-matrix representation to attain improved computational performances. The computational advantage and the model performances have been validated through a simulation study.