  Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of
order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the
adjacency matrix of $G$, respectively. In this paper, we present a sharp upper
bound for the signless spectral radius of $G$ without any tree and characterize
all extremal graphs which attain the upper bound, which may be regarded as a
spectral extremal version for the famous Erd\H{o}s-S\'{o}s conjecture.
