Solution:
Since $n$ and $n+1$ are coprime numbers, if $n(n+1)$ is a perfect square, then each of $n$ and $n+1$ has to be a perfect square itself. However, that is impossible since if $n = x^{2}$ and $n+1 = y^{2}$ we would have $1 = y^{2} - x^{2} = (y - x)(y + x)$ and $1$ can't be expressed as a product of two different integers. Hence there are no such integers.