If $Q$ reduces to $L$ then $\overline{Q}$ reduces to $\overline{L}$
By hypothesis $Q \le_p L$, i.e., there exists a poly-time computable function $f(x)$ such that $x \in L \iff f(x) \in Q$. Then: 
$$
x \in \overline{L} \iff x \not\in L \iff f(x) \not\in Q \iff f(x) \in \overline{Q}.
$$

Therefore $\overline{L} \le_p \overline{Q}$.