The log-Gaussian Cox process is a flexible and popular stochastic process for modeling point patterns exhibiting spatial and space-time dependence. Model fitting requires approximation of stochastic integrals which is implemented through discretization over the domain of interest. With fine scale discretization, inference based on Markov chain Monte Carlo is computationally burdensome because of the cost of matrix decompositions and storage, such as the Cholesky, for high dimensional covariance matrices associated with latent Gaussian variables. This article addresses these computational bottlenecks by combining two recent developments: (i) a data augmentation strategy that has been proposed for space-time Gaussian Cox processes that is based on exact Bayesian inference and does not require fine grid approximations for infinite dimensional integrals, and (ii) a recently developed family of sparsity-inducing Gaussian processes, called nearest-neighbor Gaussian processes, to avoid expensive matrix computations. Our inference is delivered within the fully model-based Bayesian paradigm and does not sacrifice the richness of traditional log-Gaussian Cox processes. We apply our method to crime event data in San Francisco and investigate the recovery of the intensity surface.