Abstract There are several models for soft regression analysis in the literature, but relatively few are based on quantiles, and these models are limited to the linear case. As quantile-based regression models offer a series of benefits (like robustness and handling of asymmetric distributions) but have not been considered in the nonlinear case, we present the first soft nonlinear quantile-based regression model in this paper. Considering nonlinearity instead of limiting to linearity in the modeling brings numerous advantages such as a higher flexibility, more accurate predictions, a better model fit and an improved explainability/interpretability of the model. In particular, we embed fuzzy quantiles into nonlinear regression analysis with crisp predictor variables and fuzzy responses. We propose a new method for parameter estimation by implementing a three-stage technique on the basis of the center and the spreads. In the framework of this procedure, we utilize kernel-fitting, a least quantile loss function, least absolute errors, and generalized cross-validation criteria to estimate the model parameters. We perform comprehensive comparative analysis with other soft nonlinear regression models that have demonstrated superiority in previous studies. The results reveal that the proposed nonlinear quantile-based regression technique leads to better outcomes compared to the competitors.