In this work we present a new construction of basis functions that generate the space of geometrically smooth splines on an unstructured quadrilateral mesh. The basis is represented in terms of biquintic Bézier polynomials on each quadrilateral face. The gluing along the face boundaries is achieved using quadratic gluing data functions, leading to globally G1-smooth spaces. We analyze the latter space and provide a combinatorial formula for its dimension as well as an explicit basis construction. Moreover, we assess the use of this basis in point cloud fitting problems. To apply G1 least squares fitting, a quadrilateral structure as well as parameters in each quadrilateral is required. Even though the general problem of segmenting and parametrizing point clouds is beyond the focus of the present work, we describe a procedure that produces such a structure as well as patch-local parameters. Our experiments demonstrate the accuracy and smoothness of the obtained reconstructed models in several challenging instances.