Abstract: To better capture the characteristics of asymmetry and structural fluctuations observed in count time series, this study delves into the application of the quantile regression (QR) method for analyzing and forecasting nonlinear integer-valued time series exhibiting a piecewise phenomenon. Specifically, we focus on the parameter estimation in the first-order Self-Exciting Threshold Integer-valued Autoregressive (SETINAR(2,1)) process with symmetry, asymmetry, and contaminated innovations. We establish the asymptotic properties of the estimator under certain regularity conditions. Monte Carlo simulations demonstrate the superior performance of the QR method compared to the conditional least squares (CLS) approach. Furthermore, we validate the robustness of the proposed method through empirical quantile regression estimation and forecasting for larceny incidents and CAD drug call counts in Pittsburgh, showcasing its effectiveness across diverse levels of data heterogeneity.