Butler \cite{But1} constructed simultaneous cospectral graphs for the adjacency and normalized Laplacian matrices, and asked the same for all three matrices, namely, adjacency, Laplacian and normalized Laplacian. In this paper, we determine the full adjacency, Laplacian and normalized Laplacian spectrum of the $Q$-vertex join and $Q$-edge join of a connected regular graph with an arbitrary regular graph in terms of their respective eigenvalues. Applying these results we construct some non-regular $A$-cospectral, $L$-cospectral and $\mathcal{L}$-cospectral graphs which gives a partial answer of the question asked by Butler \cite{But1}. Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.