We explicitly construct the canonical rational models of Shimuracurves, both analytically in terms of modular forms andalgebraically in terms of coefficients of genus 2 curves, in thecases of quaternion algebras of discriminant 6 and 10. Thisemulates the classical construction in the elliptic curvecase. We also give families of genus 2 QM curves, whose Jacobiansare the corresponding abelian surfaces on the Shimura curve, andwith coefficients that are modular forms of weight 12. We applythese results to show that our j-functions are supported exactlyat those primes where the genus 2 curve does not admitpotentially good reduction, and construct fields where thispotentially good reduction is attained. Finally, using j, weconstruct the fields of moduli and definition for some moduliproblems associated to the Atkin-Lehner group actions