  To a first approximation, the quark mixing matrix has $\theta^q_{13} =
\theta^q_{23} = 0$, whereas the lepton mixing matrix has $\theta^l_{23} =
\pi/4$. We show how this structure may be understood if the family symmetry is
$Q_8$, the quaternion group of eight elements. We find three viable scenarios
for the Majorana neutrino mass matrix, each depending on 4 parameters and
predicting a specific mass spectrum. The phenomenology of the two Higgs
doublets which generate the Yukawa sector is analyzed and testable predictions
are derived. We discuss also the closely related model based on $D_4$, the
symmetry group of the square.
