From answers here for any given integer $c$ we can tell if there exists $a$ and $b$ so that $c^2=a^2+b^2$ if and only if we can find prime factors of $c$ and check if any of them is congruent to $1$ mod $4$.
Now the question is from all prime factors of $c$ that are congruent to $1$ mod $4$ can we tell how many distinct pairs of $a_i, b_i$ exist?
As an example $c=65=5 \times 13$ and it has $4$ possible triangles that are:
$[16, 63, 65], [25, 60, 65], [33, 56, 65], [39, 52, 65]$