Summary: We show that any abelian surface with multiplication by the quaternion \(\mathbb Q\)-algebra of discriminant 6, with field of moduli \(\mathbb Q\) and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining complex multiplication points in characteristic 0 and \(p\) and the values of a certain \(j\)-function on the associated moduli space at these points.