In the present framework, the q-homotopy analysis transform method (q-HATM) we find the solution for the equation describing the interaction between Langmuir waves and the ion-acoustic waves in the plasma, called Klein-Gordon-Zakharov equations. The projected solution procedure is elegant unification of q-homotopy analysis technique with Laplace transform. The considered coupled nonlinear system analyzed with the different fractional-order to ensure the efficiency and applicability of the hired solution procedure. For distinct values of the parameters offered by the considered method and for different fractional-order, the physical behaviors of achieved solutions are captured in terms of surface and contour plots. To illustrate the accuracy and reliability of the considered method, we conducted the numerical simulation for different fractional order with the change in variables. The obtained results show that, the projected scheme is highly methodical, very effective, easy to implement and accurate to investigate the nature of fractional nonlinear coupled system exemplifying the real-world problems.