Abstract: To address variable dependence and evaluate quantile-specific effects by covariates, spatial quantile autoregressive (SQAR) models have been introduced. Conventional quantile regression focuses solely on the fitting models but ignores the examinations of multiple conditional quantile functions, which provides a comprehensive view of the relationship between the response and covariates. Thus, it is necessary to investigate the different regression slopes at different quantiles, especially in situations where the quantile coeffcients share some common feature. However, traditional Wald multiple tests not only increase the burden of computation but also leads to a higher False Discovery Rate (FDR). In this paper, we transform the estimation and examination problem into a penalization problem, which estimates the parameters at different quantiles and identifies the interquantile commonality at the same time. Based on instrumental variables and shrinkage techniques, we propose a Two-stage Interquantile Estimation (TSIE) method, including Fused Adaptive LASSO (FAL) and Fused Adaptive Sup-norm (FAS) estimators. The oracle properties of the TSIE method were established. Through Monte Carlo simulations and numerical investigations of Crime data, it is demonstrated that the proposed method leads to higher estimation effciency than the traditional quantile regression.