  Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite
digraphs on $n$ vertices which contain a complete bipartite subdigraph
$\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q
\leq n$. In this paper, we study the Q-index (i.e. the signless Laplacian
spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph
that has the minimum Q-index.
