In this paper we present the numerical evolution of a test scalar field on a Minkowski background using adaptive mesh refinement techniques (AMR). The Dynamics of the scalar field is given by the Klein Gordon equation with an exponential potential, which has been used as a model of quintessence scalar fields. As a first step in this work a description of the AMR algorithm is presented. Then we perform an analysis related to the convergence of the numerical simulations, founding convergence of second order, which is consistent with the second order finite difference scheme used.