Let $ABCD$ be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:

$A(a, 0)$, $B(0, b)$, $C(-c, 0)$, $D(0, -d)$, where $1 \le a, b, c, d \le m$ and $a, b, c, d, m$ are integers.

It can be shown that for $m = 4$ there are exactly $256$ valid ways to construct $ABCD$. Of these $256$ quadrilaterals, $42$ of them strictly contain a square number of lattice points.

How many quadrilaterals $ABCD$ strictly contain a square number of lattice points for $m = 100$?