We describe a method to determine all the isomorphism classes of
principal polarizations of the modular abelian surfaces $A_f$ with
quaternionic multiplication attached to a normalized newform $f$
without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$
whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.