This paper investigates a fully Bayesian way to solve the simultaneous localization and spatial prediction (SLAP) problem using a Gaussian Markov random field (GMRF) model. The objective is to simultaneously localize robotic sensors and predict a spatial field of interest using sequentially obtained noisy observations collected by robotic sensors. The set of observations consists of the observed uncertain poses of robotic sensing vehicles and noisy measurements of a spatial field. To be flexible, the spatial field of interest is modeled by a GMRF with uncertain hyperparameters. We derive an approximate Bayesian solution to the problem of computing the predictive inferences of the GMRF and the localization, taking into account observations, uncertain hyperparameters, measurement noise, kinematics of robotic sensors, and uncertain localization. The effectiveness of the proposed algorithm is illustrated by simulation results.