I will expand my comment into an answer.
First, any automorphism of $Q$ is induced by an automorphism of the ambient projective space. This is because any automorphism of $Q$ must fix the canonical bundle, which here is $\mathcal{O}_Q(-3)$. In particular any automorphism must fix $\mathcal{O}_Q(1)$, hence is an automorphism of the ambient projective space.
The collection of projective automorphisms which preserve the quadric is exactly $PO(Q)$. The result then follows the fact that $PO(Q) \cong PO(5)$, together with the fact that the natural map $SO(2k+1)\to PO(2k+1)$ is an isomorphism, for any natural number $k$ (see for instance wikipedia.)