Find odd primes $p$ and $q$ such that $(p-1)\mid {3q-1}$ and $(q-1)\mid{3p-1}$.

The condition tells us that $\frac{3p-1}{q-1}$,$\frac{3q-1}{p-1}$, and $\frac{3p-1}{p-1} \frac{3q-1}{q-1}$ is an integer. However, for $p,q\ge5$, we have

$$\frac{3p-1}{p-1} \frac{3q-1}{q-1} < \frac{3p}{\frac{4}{5}p} \times \frac{3q}{\frac{4}{5}q} <15$$

Here's a hint: When you have a problem like this, with two separate divisibilities, you want to make the right sides look the same. In particular, here $$p-1|3q-1+3(p-1)=3p+3q-4,$$ and $q-1$ must also divide this by symmetry. As a result, $$\operatorname{lcm}(p-1,q-1)|3p+3q-4.$$ The left side should be larger than the right side for most $(p,q)$ as long as $p-1$ and $q-1$ can't share big factors, which would allow you to finish; can you determine whether $p-1$ and $q-1$ can share large factors?

Let $3q-1 = k(p-1)$. Try $k=1,2,3,\ldots$ until you can show $k$ is too large.