Summary: We study the Dirichlet problem for the quasilinear second-order sub-elliptic equation \[ \sum^m_{i,j= 1} X^*_i(A_{i,j}(x, u)X_j u)+ \sum^m_{j= 1} B_j(x, u) X_j u+ C(x, u)= 0 \quad \text{in }\Omega,\quad u=\varphi\quad \text{on } \partial\Omega, \] where \(X= \{X_1,\dots, X_m\}\) is a system of real smooth vector fields which satisfies Hörmander's condition, \(A_{i,j}, B_j, C\in C^\infty(\overline\Omega\times \mathbb{R})\) and \((A_{i,j}(x, z))\) is a positive definite matrix. We prove the existence and the maximal regularity of solutions in the ``non-isotropic'' Hölder space associated with the system of vector fields \(X\).