Solution:
Solving the following system of equations using the elimination method:
$$\left\{\begin{array}{r} x^{2}+y=12 \\ x+y=12 \end{array}\right.$$
we get the ordered pairs $(0,12)$ and $(1,11)$. That is, the given graphs intersect at $(0,12)$ and $(1,11)$, whose distance is $\sqrt{(0-1)^{2}+(12-11)^{2}}=\sqrt{2}$.