  We construct one dimensional families of Abelian surfaces with quaternionic
multiplication which also have an automorphism of order three or four. Using
Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we
find the Shimura curve parametrizing these Abelian surfaces in a specific case.
We explicitly relate these surfaces to the Jacobians of genus two curves
studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in
Barth's moduli space which parametrizes Abelian surfaces with real
multiplication by Z[\sqrt{2}].
