We obtain estimates for the number of integral solutions in large balls, of inequalities of the form |Q(x, y)| < ϵ, where Q is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which Q vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.