  A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In
this paper, we determine the maximal signless Laplacian spectral radius and the
corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order
$n$. Furthermore, we determine the first three unicyclic, and bicyclic,
$C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal.
Similar results are obtained for the (combinatorial) Laplacian.
