Assume that there is $k$ such that $3^m + 3^n + 1 = k^2$. Obviously, $k$ is odd. The last equation is equivalent to $3^m + 3^n = k^2 - 1$.

It is easy to see that for odd number $k$, $8$ divides $k^2 - 1$. Since powers of $3$ are congruent to $1$ or $3$ modulo $8$, the number $3^m + 3^n$ is congruent to $2$, $4$ or $6$ modulo $8$. Therefore $3^m + 3^n = k^2 - 1$ leads to a contradiction.

There are no positive integers $m$ and $n$ such that $3^m + 3^n + 1$ is a perfect square.