Example 1 Find all prime numbers $p, q, r (p \leqslant q \leqslant r)$ such that
$$p q+r, p q+r^{2}, q r+p, q r+p^{2}, r p+q, r p+q^{2}$$

are all prime numbers.
If $p>2$, then $p, q, r$ are all odd numbers, in which case $pq+r$ is an even number greater than 2, leading to a contradiction. Therefore, $p=2$. Now, the numbers
$$2q+r, 2q+r^2, qr+2, qr+4, 2r+q, 2r+q^2$$

are all prime numbers.
If $q, r$ include an even number, then $qr+2$ is an even number greater than 2, leading to a contradiction. Therefore, $q, r$ are both odd prime numbers. If $q>3$, then $3 \nmid qr$. In this case, if $qr \equiv 1 \pmod{3}$, then $qr+2 \equiv 0 \pmod{3}$, contradicting the fact that $qr+2$ is a prime number; if $qr \equiv 2 \pmod{3}$, then $qr+4 \equiv 0 \pmod{3}$, contradicting the fact that $qr+4$ is a prime number. Therefore, $q=3$. Thus, the numbers
$$6+r, 6+r^2, 3r+2, 3r+4, 2r+3, 2r+9$$

are all prime numbers.
If $r \neq 5$, then $r \neq 0 \pmod{5}$. However, when $r \equiv 1, 2, 3, 4 \pmod{5}$, the corresponding numbers $3r+2, 3r+4, 2r+9, 6+r$ are multiples of 5, leading to a contradiction. Therefore, $r=5$.

Direct verification shows that they satisfy the conditions. The required prime numbers are
$$p=2, q=3, r=5$$