Relevant information from networked systems can be obtained by analyzing the spectra of matrices associated to their graph representations. In particular, the eigenvalues and eigenvectors of the Markov matrix and related Laplacian and normalized Laplacian matrices allow the study of structural and dynamical aspects of a network, like its synchronizability and random walks properties. In this study we obtain, in a recursive way, the spectra of Markov matrices of a family of rotationally invariant weighted Sierpinski graphs. From them we find analytic expressions for the weighted count of spanning trees and the random target access time for random walks on this family of weighted graphs.