Answer: 117

Small Hint:

The left side factors as $(3x + 2y)(4x - 3y)$

Big Hint:

One line gives $(x, y) = (3t, 4t)$ with $|t| \le 25;$ the other gives $(2t, -3t)$ with $|t| \le 33.$ Don’t count the origin twice.

Solution:

The equation factors as $$\begin{gathered} 12x^2 - xy - 6y^2 \\ = (3x + 2y)(4x - 3y) \\ = 0, \end{gathered}$$ so every solution has $4x = 3y$ or $3x = -2y.$

Integer solutions of $4x = 3y$ are $(x, y) = (3t, 4t);$ the constraint $|4t| \le 100$ gives $-25 \le t \le 25,$ or $51$ pairs. Integer solutions of $3x = -2y$ are $(x, y) = (2t, -3t);$ the constraint $|3t| \le 100$ gives $-33 \le t \le 33,$ or $67$ pairs. The families overlap only at $(0, 0),$ so the count is $51 + 67 - 1 = 117.$