Recently, zero-inflated Gaussian processes (GPs) have been proposed as probabilistic machine learning models for observed spatio-temporal data that contain many close-to-zero entries. In this work, we extend zero-inflated GPs to sparse count data via the zero-inflated Poisson likelihood. This change no longer admits a closed-form computation of the training objective, so we use automatic differentiation variational inference to perform approximate posterior estimation. Our motivating application is the prediction of the number of opioid-related overdose deaths that will occur in the next 3 months in each of 1620 census tracts across the state of Massachusetts, given historical decedent data and socio-economic covariates. We find zero-inflated GPs can prioritize regions in need of near-term public health interventions better than alternative models at finer spatial and temporal resolutions than most prior efforts. Surprisingly, we find that this model is successful even when using Normal likelihoods instead of the zero-inflated Poisson.