The empirical best linear unbiased prediction (EBLUP) method has been the dominant model-based approach in small area estimation. As an alternative to this frequentist method, the observed best prediction (OBP) method, also frequentist, was proposed by Jiang et al. (2011) where the parameters of the model are estimated by minimizing an objective function which is implied by the total mean squared prediction error. In a recent article, Datta et al. (2023) followed a general Bayesian approach, proposed recently by Bissiri et al. (2016), to develop a quasi-Bayesian method by appropriately calibrating the objective function for the OBP method for the Fay-Herriot model. In a different article, Chung and Datta (2022) demonstrated that in the absence of covariates with good predictive power the small area estimates from the standard Fay-Herriot model can be improved by using spatially dependent random effects. In this article, we develop a quasi-Bayesian small area estimation method using several spatial alternatives to the independent Fay-Herriot random effects model. Evaluation of the proposed method based on an application to estimation of four-person family median incomes for the U.S. states shows its usefulness. Limited but related simulation studies for the median incomes application reinforce our conclusion.