5. As shown in Figure 1, a circle centered at point $M$ on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ is tangent to the $x$-axis at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\triangle M P Q$ is an equilateral triangle, then the eccentricity of the hyperbola is $\qquad$
5. $\sqrt{3}$.

Since $\odot M$ is tangent to the $x$-axis at the focus $F$ of the hyperbola, we have $|M F|=\frac{b^{2}}{a}$, and the distance from point $M$ to the $y$-axis is $c$.

Thus, the side length of the equilateral $\triangle M P Q$ is $\frac{b^{2}}{a}$, and the height from side $P Q$ is $c$.
From $c=\frac{\sqrt{3}}{2} \cdot \frac{b^{2}}{a} \Rightarrow 2 a c=\sqrt{3}\left(c^{2}-a^{2}\right)$.
Solving this, we get $\frac{c}{a}=\sqrt{3}$, which means the eccentricity of the hyperbola is $\sqrt{3}$.