Q-Morph: An indirect approach to advancing front quad meshing. $Q$-Morph is a new algorithm for generating all-quadrilateral meshes on bounded three-dimensional surfaces. After first triangulating the surface, the triangles are systematically transformed to create an all-quadrilateral mesh. An advancing front algorithm determines the sequence of triangle transformations. Quadrilaterals are formed by using existing edges in the triangulation, by inserting additional nodes, or by performing local transformations to the triangles. A method, typically used for recovering the boundary of a Delaunay mesh, is used on interior triangles to recover quadrilateral edges. Any number of triangles may be merged to form a single quadrilateral. Topological clean-up and smoothing are used to improve final element quality. $Q$-Morph generates well-aligned rows of quadrilaterals parallel to the boundary of the domain while maintaining a limited number of irregular internal nodes. The proposed method also offers the advantage of avoiding expensive intersection calculations commonly associated with advancing front procedures. A series of examples of $Q$-Morph meshes are also presented to demonstrate the versatility of the proposed method.