  This note presents a new spectral version of the graph Zarankiewicz problem:
How large can be the maximum eigenvalue of the signless Laplacian of a graph of
order $n$ that does not contain a specified complete bipartite subgraph. A
conjecture is stated about general complete bipartite graphs, which is proved
for infinitely many cases.
  More precisely, it is shown that if $G$ is a graph of order $n,$ with no
subgraph isomorphic to $K_{2,s+1},$ then the largest eigenvalue $q(G)$ of the
signless Laplacian of $G$ satisfies \[
q(G)\leq\frac{n+2s}{2}+\frac{1}{2}\sqrt{(n-2s)^{2}+8s}, \] with equality
holding if and only if $G$ is a join of $K_{1}$ and an $s$-regular graph of
order $n-1.$
