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Task functional magnetic resonance imaging (fMRI) is a type of neuroimaging data used to identify areas of the brain that activate during specific tasks or stimuli. These data are conventionally modeled using a massive univariate approach across all data locations, which ignores spatial dependence at the cost of model power. We previously developed and validated a spatial Bayesian model leveraging dependencies along the cortical surface of the brain in order to improve accuracy and power. This model utilizes stochastic partial differential equation spatial priors with sparse precision matrices to allow for appropriate modeling of spatially-dependent activations seen in the neuroimaging literature, resulting in substantial increases in model power. Our original implementation relies on the computational efficiencies of the integrated nested Laplace approximation (INLA) to overcome the computational challenges of analyzing high-dimensional fMRI data while avoiding issues associated with variational Bayes implementations. However, this requires significant memory resources, extra software, and software licenses to run. In this article, we develop an exact Bayesian analysis method for the general linear model, employing an efficient expectation-maximization algorithm to find maximum a posteriori estimates of task-based regressors on cortical surface fMRI data. Through an extensive simulation study of cortical surface-based fMRI data, we compare our proposed method to the existing INLA implementation, as well as a conventional massive univariate approach employing ad-hoc spatial smoothing. We also apply the method to task fMRI data from the Human Connectome Project and show that our proposed implementation produces similar results to the validated INLA implementation. Both the INLA and EM-based implementations are available through our open-source BayesfMRI R package.