Answer: 501

Small Hint:

Factor: $a^2 - b^2 + c^2 - d^2$ $= (a-b)(a+b)$ $+ (c-d)(c+d)$ $\ge (a+b) + (c+d)$

Big Hint:

Equality forces $a - b = c - d = 1,$ so $a + c = 1006;$ then find the range of $a$ allowed by $b \gt c$ and $d \ge 1$

Solution:

Factoring, $$\begin{gathered} a^2 - b^2 + c^2 - d^2 \\ = (a-b)(a+b) \\ {}+ (c-d)(c+d) \\ \ge (a+b) + (c+d) = 2010, \end{gathered}$$ since $a - b \ge 1$ and $c - d \ge 1.$ Equality holds, so $a - b = c - d = 1,$ that is, $b = a - 1$ and $d = c - 1.$ Then $2010 = a + (a-1) + c + (c-1)$ gives $a + c = 1006.$

The condition $b \gt c$ means $a - 1 \gt c = 1006 - a,$ so $a \ge 504,$ and $d \ge 1$ means $c \ge 2,$ so $a \le 1004.$ Every $a$ in this range works, via $(a, b, c, d)$ $= (a,\, a-1,\,$ $1006-a,\, 1005-a).$

The count is $1004 - 504 + 1 = 501.$