An approach for direct discretization of 3D surfaces into mixed quad-dominant meshes using a triangular mesh generator has been outlined in this paper. The octree is used to control the mesh gradation and to efficiently perform spatial localization. To make the algorithm capable of dealing with a wide variety of surfaces, the class of tensor product polynomial surfaces has been considered. The feedback to the parametric space of these surfaces is used to efficiently resolve some local problems. The actual discretization is split into two phases. Initially, a mixed mesh is created using an advancing front based mesh generator. The quadrilateral elements are formed by merging two adjacent triangles consecutively created by the triangular kernel. Since the merging is not always possible, a relatively large number of triangular elements is present in the initial mesh. Most of these triangles are removed from the mesh during the cleanup process performed in the second phase. The cleanup process consists of several times repeated Laplacian smoothing combined with topological transformations. The proposed strategy is capable to produce uniform and graded mixed meshes of high quality and its performance has been demonstrated on several examples. Note that the strategy can be easily used to generate all-quadrilateral meshes if the final mixed quad-dominant mesh is subjected to one-level refinement.