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Yes, it trivially happens if $G = \langle g \rangle$ is the cyclic group of order $pq$ -- then all $3$ conjugacy classes mentioned have precisely one element. A smallest example of a nonabelian group $G$ having such an element is $$ G \ = \ \langle (1,3,2)(4,5)(6,7), (2,3)(5,7) \rangle, \ \ g \ = \ (1,2,3)(4,5)(6,7). $$ Then $|G| = 24$, and we have $|(g^2)^G| = |(g^3)^G| = |(g^{3-2})^G| = 2$.