In this paper, we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps, by means of the generation of a unit mesh with respect to a riemannian structure. We propose then an automatic triangular to quadrilateral mesh conversion scheme, which generalizes the standard case to the anisotropic context. In addition, we introduce an optimal vertex smoothing procedure. Application test examples, in particular a CFD test, are given to demonstrate the efficiency of the proposed method.