If $n$ is even then $n^n$ is always perfect square. $n=2,4,6,\dots,2020$. If $n$ is odd then $n$ must be perfect square. $n=1^2,3^2,5^2,\dots, 43^2$ Thus, there are $1010+22=1032$ integers $n$ such that $n^n$ is perfect square.

The answer is 1011, I think so.