Call a quadruple of positive integers $(a, b, c, d)$ fruitful if there are infinitely many integers $m$ such that $\text{gcd} (am + b, cm + d) = 2019$. Find all possible values of $|ad-bc|$ over fruitful quadruples $(a, b, c, d)$.
1. Consider an arbitrary fruitful quadruple \((a, b, c, d)\). Since there are infinitely many integers \(m\) such that \(\gcd(am + b, cm + d) = 2019\), we can write:
   \[
   \gcd(am + b, cm + d) = 2019
   \]
   This implies that both \(am + b\) and \(cm + d\) are multiples of 2019 for infinitely many \(m\).

2. We can express this condition as:
   \[
   am + b = 2019k \quad \text{and} \quad cm + d = 2019l
   \]
   for some integers \(k\) and \(l\).

3. Consider the expression:
   \[
   |c(am + b) - a(cm + d)|
   \]
   Substituting the values from the previous step, we get:
   \[
   |c(2019k) - a(2019l)| = 2019|ck - al|
   \]

4. Since \(k\) and \(l\) can take infinitely many values, the expression \(2019|ck - al|\) must be a multiple of 2019. Therefore, we have:
   \[
   |ad - bc| = 2019|k - l|
   \]
   This shows that \(|ad - bc|\) must be a nonnegative multiple of 2019.

5. To show that this is a sufficient condition, consider the quadruples \((a, b, c, d) = (673, 673x, 3, 3y)\). For these values, we have:
   \[
   |ad - bc| = |673 \cdot 3y - 3 \cdot 673x| = 2019|x - y|
   \]

6. We need to show that \(|x - y|\) can attain any nonnegative integer value. This means we need to show that \(3p - 673q\) can attain every integer value for some pair of relatively prime integers \(p\) and \(q\).

7. Consider the following cases:
   - For \(k\) congruent to 2 modulo 3, set \(q = 1\) and \(p = \frac{k + 673}{3}\). These values generate a pair of relatively prime integers.
   - For \(k\) congruent to 1 modulo 3, set \(q = -1\) and \(p = \frac{k - 673}{3}\). These values are also relatively prime.
   - For multiples of 3, let \(3p - 673q = 3k_1\). Clearly, \(q\) must be a multiple of 3, so let \(q = 3q_1\). This transforms the equation to:
     \[
     3p - 2019q_1 = 3k_1 \implies p = 673q_1 + k_1
     \]
     and \(\gcd(p, 3q_1) = 1\).

8. For any value of \(k_1\) that is not a multiple of 673, Dirichlet's Theorem guarantees that there will exist a prime value of \(p > 10^{10}\) and an integer value of \(q_1\) that satisfies the equation. Since this value of \(p\) is much greater than \(3q_1 = q\), it must be relatively prime to \(q\), and thus it satisfies our equations.

9. For \(k_1\) that is a multiple of 673 (i.e., \(k_1 = 673k_2\)), substituting this in gives:
   \[
   p = 673(q_1 + k_2)
   \]
   Therefore, \(p\) must be a multiple of 673, or \(p = 673p_1\). Substituting this in gives:
   \[
   p_1 = q_1 + k_2 \implies p_1 - q_1 = k_2
   \]
   and \(\gcd(673p_1, 3q_1) = 1\).

10. If \(k_2\) is congruent to 0 or 1 modulo 3, then letting \(q_1 = 1\) gives a solution. Note that the corresponding value of \(p_1\) cannot be a multiple of 3, so:
    \[
    \gcd(673p_1, 3q_1) = \gcd(673p_1, 3) = 1
    \]
    and this construction works. If \(k_2\) is congruent to 2 modulo 3, then letting \(q_1 = -1\) gives a solution. Note that the corresponding value of \(p_1\) cannot be a multiple of 3, and the same logic follows as before.

11. Since we have provided an explicit construction or a method of existence of a solution for every possible integer, we have proven the desired lemma.

12. We have just shown that \(3p - 673q\) can attain every integer for some pair of relatively prime integers \(p\) and \(q\). Therefore, \(|x - y| = |3p - 673q|\) can attain every nonnegative integer for suitable coprime integers \(p\) and \(q\). Hence, \(|ad - bc| = 2019|x - y|\) can attain every nonnegative multiple of 2019 for a suitable fruitful quadruple \((a, b, c, d) = (673, 673x, 3, 3y)\).

13. Therefore, all of the possible values of \(|ad - bc|\) over fruitful quadruples \((a, b, c, d)\) are:
   \[
   \boxed{\text{the nonnegative multiples of 2019}}
   \]