Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).
1. Let \(a, b, c, d\) be the four integer roots of the polynomial \(x^4 + 2px^2 + qx + p^2 - 36 = 0\). By Vieta's formulas, we know:
   \[
   a + b + c + d = 0
   \]
   This implies that the sum of the roots is zero.

2. We also have:
   \[
   ab + ac + ad + bc + bd + cd = 2p
   \]
   and
   \[
   abcd = p^2 - 36
   \]

3. Since \(a + b + c + d = 0\), we can pair the roots such that \(a \equiv b \pmod{2}\) and \(c \equiv d \pmod{2}\). Without loss of generality, assume \(a^2 \geq b^2\), \(c^2 \geq d^2\), and \(ab \geq cd\).

4. We can express the polynomial in terms of its roots:
   \[
   (x - a)(x - b)(x - c)(x - d) = x^4 + 2px^2 + qx + p^2 - 36
   \]

5. From the given conditions, we have:
   \[
   ab + cd - (a + b)^2 = 2p
   \]
   and
   \[
   abcd = p^2 - 36
   \]

6. We can derive the following equation:
   \[
   (ab + cd)^2 - 2(ab + cd)(a + b)^2 + (a + b)^4 - 4abcd = 144
   \]

7. Simplifying the above equation, we get:
   \[
   (ab + cd)^2 - 2(ab + cd)(a + b)^2 + (a + b)^4 - 4abcd = (ab - cd)^2 + (a + b)^2((a + b)^2 - 2(ab + cd))
   \]

8. Since \((a + b)^2 = (c + d)^2\), we have:
   \[
   a^2 + b^2 - 2cd = c^2 + d^2 - 2ab \implies a^2 + b^2 - 2cd = \frac{(a^2 + b^2 - 2cd) + (c^2 + d^2 - 2ab)}{2} = \frac{(a - b)^2 + (c - d)^2}{2}
   \]

9. Therefore, we get:
   \[
   144 = (ab - cd)^2 + (a + b)^2 \frac{(a - b)^2 + (c - d)^2}{2} \implies 288 = 2(ab - cd)^2 + (a^2 - b^2)^2 + (c^2 - d^2)^2
   \]

10. Since \(4 \mid a^2 - b^2\) and \(4 \mid c^2 - d^2\), we have:
    \[
    16 \mid 2(ab - cd)^2 \implies 4 \mid ab - cd
    \]

11. Thus, we can write:
    \[
    18 = 2\left( \frac{ab - cd}{4} \right)^2 + \left( \frac{a^2 - b^2}{4} \right)^2 + \left( \frac{c^2 - d^2}{4} \right)^2
    \]

12. The only non-negative integer solutions \((x, y, z)\) to \(18 = 2x^2 + y^2 + z^2\) are:
    \[
    (0, 3, 3), (1, 4, 0), (1, 0, 4), (2, 3, 1), (2, 1, 3), (3, 0, 0)
    \]

13. From these solutions, we can determine the possible values for \(a, b, c, d\) and subsequently find the pairs \((p, q)\).

The final answer is \( \boxed{ (p, q) } \) pairs that satisfy the given conditions.