This paper makes further progress on the following problem: given an irreducible abelian surface \(A\) over a number field, determine explicitly all the genus 2 curves whose Jacobian is isomorphic to \(A\). The authors studied this problem for irreducible principally polarized abelian surfaces in [E. González-Jiménez, J. González and J. Guàrdia, “Computations on modular Jacobian surfaces”, Lect. Notes Comput. Sci. 2369, 189–197 (2002; Zbl 1055.11038)]. In [J. González, J. Guàrdia and V. Rotger, “Abelian surfaces of \(\text{GL}_{2}\)-type as Jacobians of curves”, Acta Arith. 116, No. 3, 263–287 (2005; Zbl 1108.14032)], they developed the theoretical results related to this problem for irreducible polarized abelian surfaces, focusing on nonprincipal polarizations. In both cases, they applied their ideas to modular abelian surfaces, since the discovery of new algorithms to describe the Jacobians of modular curves makes it possible to generate explicit examples. Unfortunately, the numerical nature of these algorithms leads only to numerically-tested examples. In the paper under review, the authors discuss the case of abelian surfaces with quaternionic multiplication, whose rich endomorphism algebra allows for the combined application of additional well-known techniques. More specifically, a method is described for determining 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. Then an example is given of a surface \(A_{f}\) with quaternionic multiplication for which the authors find numerically a curve \(C\) whose Jacobian is \(A_{f}\) up to numerical approximation, and then prove that it has quaternionic multiplication and is in fact isogenous to \(A_{f}\).