Analysis: First, fix point $Q$ on the ellipse. It is evident that $|P Q|$ is maximized when $P Q$ passes through the center $O_{1}$. Therefore, to find the maximum value of $|P Q|$, we only need to find the maximum value of $\left|O_{1} Q\right|$.
Let $Q(x, y)$, then
$$\left|O_{1} Q\right|^{2}=x^{2}+(y-4)^{2} \text {. }$$

Since $Q$ is on the ellipse, we have $\frac{x^{2}}{9}+y^{2}=1$, which means
$$x^{2}=9\left(1-y^{2}\right) \text {. }$$

Substituting equation (2) into equation (1) yields,
$$\begin{array}{l} \left|O_{1} Q\right|^{2}=9\left(1-y^{2}\right)+(y-4)^{2} \\ =-8 y^{2}-8 y+25=-8\left(y+\frac{1}{2}\right)^{2}+27 . \end{array}$$

Since point $Q$ moves on the ellipse, $-1 \leqslant y \leqslant 1$. Therefore, when $y=-\frac{1}{2}$,
$$\left|O_{1} Q\right|_{\max }=3 \sqrt{3},|P Q|_{\max }=3 \sqrt{3}+1 \text {. }$$

Note: In problems involving the analytic geometry of circles, it is often necessary to flexibly apply the relevant properties of circles.