The authors describe the graded Betti numbers occurring in the minimal graded free resolution of the homogeneous vanishing ideal of a reduced 0-dimensional scheme \(X \subseteq \mathbb{P}^3\) which is contained in a smooth quadric surface \(Q \subseteq \mathbb{P}^3\). This resolution is constructed from the knowledge of a particular type of locally free resolution of the relative ideal sheaf \(\overline {{\mathcal I}}_X\) of \(X\) in \({\mathcal O}_Q\), namely a resolution built from summands of the form \({\mathcal O}_Q (a,b)\) with bidegrees \((a,b)\) in a certain “good” rectangle. They also prove that any 0-dimensional subscheme of \(Q\) is determinantal, and that the Hilbert function of a generic set of points on \(Q\) determines its graded Betti numbers which are exactly given by the fourth difference function of the Hilbert function of \(X\). The paper is part of a programme outlined by the authors previously [see S. Giuffrida, R. Maggioni and A. Ragusa in: zero-dimensional schemes, Proc. Conf., Ravello 1992, 191-204 (1994; Zbl 0826.14029)].