Graphical models with latent count variables arise in a number of fields. Standard exact inference techniques such as variable elimination and belief propagation do not apply to these models because the latent variables have countably infinite support. As a result, approximations such as truncation or MCMC are employed. We present the first exact inference algorithms for a class of models with latent count variables by developing a novel representation of countably infinite factors as probability generating functions, and then performing variable elimination with generating functions. Our approach is exact, runs in pseudo-polynomial time, and is much faster than existing approximate techniques. It leads to better parameter estimates for problems in population ecology by avoiding error introduced by approximate likelihood computations.