<p>The inherent nonsmoothness of quantile regression problems creates significant computational difficulties. These challenges are further exacerbated when an additional nonconvex and nonsmooth sparsity penalty is incorporated into the model. By leveraging the Moreau envelope method to smooth out inherent nonsmooth components, we derive a stepwise smooth approximation scheme tailored for nonsmooth nonconvex sparsity-penalized quantile regression. However, this smoothing process introduces severe ill-conditioning into the problem formulation. To resolve this issue and nonconvexity due to the folded concave penalty, we propose an innovative two-metric variable scaled splitting algorithm that synergizes second-order optimization with first-order information from nonconvex penalties. The algorithm comprises a forward step that uses a quasi-Newton strategy to modify the descent direction and a backward step that employs an adaptive weight-metric matrix for the construction of the proximal operator associated with the nonconvex penalties. Theoretical analysis demonstrates the global convergence properties of the algorithm. Numerical experiments show that the proposed method outperforms widely used language packages <i>R</i> in solving penalized quantile regression problems, achieving superior computational efficiency and enhanced variable selection capabilities.</p>