If you apply a function to a random variable you get a new random variable (the function should be measurable for this to hold). Then you can proceed to calculate correlation in the usual manner. So if you take any $f$ and $g$ and normal variable $X$, the desired correlation is
Var(g(X))& = E[g^2(X)]-E[g(X)]^2
so you need to calculate 5 integrals: $E[(f\cdot g)(X)]$, $E[f(X)]$, $E[f^2(X)]$, $E[g(X)]$, $E[g^2(X)]$ to get the correlation. Even if the $X$ distribution is known and normal, there is no closed form solution for general $f$ and $g$. The particular case of $f(x)=x$ and $g(x)=x^2$ is covered by the answer of Dilip Sarwate. Since you can find closed formulas for moments of normal variables, it would be possible to derive the closed formula in the case of polynomial $f$ and $g$.