Product of Two Triangular Numbers to make Square

Theorem
Let $T_n$ be a triangular number.

Then there is an infinite number of $m \in \Z_{>0}$ such that $T_n \times T_m$ is a square number.

Proof
Since $n^2 < n \paren {n + 1} < \paren {n + 1}^2$, $n \paren {n + 1}$ cannot be a square number.

Thus there are infinitely many distinct integer solutions to the Pell's Equation:
 * $x^2 - n \paren {n + 1} y^2 = 1$

and for each solution:

Hence the result.