We consider a fusion sensor-network architecture where sensor-processor elements (nodes) observe a spatially correlated random field within a region of interest and transmit quantized observations to a fusion center. The fusion center provides feedback by broadcasting summary information to the nodes. We assume that the observations follow a linear-regression model with known field correlations and propose a Bayesian framework for adaptive quantization, fusion-center feedback, and estimation of the field and its parameters. We consider local quantile and Lloyd-Max quantizers at the nodes; both quantization schemes are based on approximate predictive measurement distributions, constructed using the feedback information from the fusion center. We also apply our estimation approach to the no-feedback scenario and present numerical examples demonstrating the performance of the proposed methods.