Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in any Hecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic $$\mathbb{Z}[\lambda]$$-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic $$\mathbb{Z}[\lambda]$$-binary quadratic forms. Keywords:Hecke groups, conjugacy class, quadratic formsCategories:11F06, 11E16, 11A55