Quantile regression (QR) has been widely studied in statistics and econometrics. However, there is no much work on nonlinear QR for vector time series. Therefore, we propose a local spatial QR method to estimate the functional-coefficient matrices of multivariate time series. We propose a local spatial quantile regression estimator (LSQR) using spatial QR and local smoothing. To improve the performance, we propose a weighted composite LSQR estimator (WCLSQR) which uses the idea of weighted composite QR. We establish the asymptotic normality of the proposed estimators, which is further used to select an optimal bandwidth and optimal weights for the estimation. Furthermore, to achieve computational efficiency, we propose a smoothed spatial QR which simplifies and accelerates the minimization problem in the spatial QR. Based on the smoothed spatial QR, we propose the smoothed LSQR and WCLSQR estimators using the same techniques as LSQR and WCLSQR. By establishing the asymptotic normality of the proposed estimators, we show that the estimators using the smoothed spatial QR can achieve comparable performance with a proper choice of the smoothing parameter while consuming much less computing resources. Simulation studies of the proposed estimators demonstrate good finite sample performance and computational efficiency. Real-world applications are also demonstrated.