Let $n+23= m^2$ and $n-30=k^2$, then we have $$(n+23)-(n-30)=m^2 - k^2 \implies 53 = (m+k)(m-k)$$

Because $53$ is a prime number which can only be factorized as $1\times 53$, Hence we must have $$m-k = 1, m+k=53\implies k=26, m=27$$

It follows that $n=m^2 -23 = \boxed{706}$.