  In this note, we present a structural description of certain connected
cographs having $k \geq 2$ main signless Laplacian eigenvalues. This result
allows us to characterize the cographs which are quasi-threshold graphs with
two main $\mathbf{Q}$-eigenvalues. In addition, we describe all the
quasi-threshold graphs belonging to the subclass of generalized core-satellite
graphs with $k \geq 2$ main $\mathbf{Q}$-eigenvalues.
