Using the theory of Higgs bundles and their stabitlity properties associated to fibered surfaces and the Viehweg-Zuo characterization of Shimura curves in the moduli space of abelian varieties in terms of Higgs bundles, we prove that there does not exist any non-compact Shimura curves in the Prym locus of totally ramified $\Z_{2p}$- or $\Z_{2p}\times (\Z_{p})^{m-1}$-covers of the projective line in $A_{g}$ for $g\geq 8$, where $p\geq 5$ is a prime number.