<p>In this work, we establish the existence of positive solutions for a class of quasilinear Schrödinger systems defined on the Heisenberg group, a non-Euclidean geometric setting that arises naturally in mathematical physics and conformal geometry. The systems under consideration are driven by quasilinear subelliptic operators of <i>Q</i>-Laplacian type and involve Schrödinger nonlinearities combining concave power-type terms with exponential growth. Depending on the involved parameters, the nonlinear terms may exhibit subcritical, critical, or supercritical behavior within the framework of the Trudinger–Moser inequality adapted to the Heisenberg group. Due to the quasilinear structure of the operator and the presence of exponential nonlinearities, standard variational methods cannot be applied directly. To overcome this difficulty, we introduce a suitable change of variables that transforms the original quasilinear system into an equivalent semilinear problem. The existence of positive solutions for the transformed system is established through a refined approximation scheme based on the Galerkin method, combined with a modified fixed point argument. Special attention is devoted to the treatment of the exponential nonlinearities and to the justification of the compactness arguments required for the passage to the limit in the approximation scheme. The results presented in this paper address a relatively unexplored class of quasilinear Schrödinger systems in subelliptic settings and provide new analytical insights into problems involving exponential nonlinearities on the Heisenberg group.</p>