In this paper, a new generalization of Bézier curves with one shape parameter is constructed. It is based on the Lupaş q-analogue of Bernstein operator, which is the first generalized Bernstein operator based on the q-calculus. The new curves have some properties similar to classical Bézier curves. Moreover, we establish degree evaluation and de Casteljau algorithms for the generalization. Furthermore, we construct the corresponding tensor product surfaces over the rectangular domain, and study the properties of the surfaces, as well as the degree evaluation and de Casteljau algorithms. Compared with q-Bézier curves and surfaces based on Phillips q-Bernstein polynomials, our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces. The shape parameters provide more convenience for the curve and surface modeling.