This can happen in a non-Abelian group for any choice of primes $p < q.$ Take a prime $r \equiv 1$ (mod $pq$), and let $G$ be a Frobenius group of order $pqr$ with kernel $K$ of order $r$ and complement $H$ of order $pq$ ( this can be done as as a cyclic group of order $r$ has a cyclic automorphism group of order $r-1$ and each non-identity automorphism fixes no non-identity element). Let $g$ be a generator of $H$. Then $C_{G}(g^{i}) = H$ for $1 \leq i < pq$,so that the conjugacy classes of $g^{p},g^{q}$ and $g^{q-p}$ all have size $r$.

This can happen in a non-Abelian group for any choice of primes $p < q.$ Take a prime $r \equiv 1$ (mod $pq$), and let $G$ be a Frobenius group of order $pqr$ with kernel $K$ of order $r$ and cyclic complement $H$ of order $pq$ ( this can be done as as a cyclic group of order $r$ has a cyclic automorphism group of order $r-1$ and each non-identity automorphism fixes no non-identity element). Let $g$ be a generator of $H$. Then $C_{G}(g^{i}) = H$ for $1 \leq i < pq$,so that the conjugacy classes of $g^{p},g^{q}$ and $g^{q-p}$ all have size $r$.