Transforming the given equation, we have
\[
2q = (p-q)\left((p-q)^2 - 1\right).
\]Since \( p \ne q \), \( p-q \) and \( q \) are coprime, so \( p-q \) must be a divisor of \( 2 \).
Here, we consider the possible cases:
- When \( p-q = 1 \), it follows that \( 2q = 0 \), which is invalid.
- When \( p-q = 2 \), substituting gives \( 2q = 6 \), which leads to \( (p, q) = (5, 3) \).
Thus, we conclude that \( (p, q) = (5, 3) \).