Any morphism from a group $G$ to an abelian group factors through the abelianized $G^{\text{ab}}$. Moreover $G$ finite implies that $G^{\text{ab}}$ is aproduct of cyclic groups and since $\hom(C_1\times C_2,-)\simeq\hom(C_1,-)\times\hom(C_2,-)$ we may assume that $G$ is finite cyclic. On the other hand $ \hom(\Bbb Z/n\Bbb Z,\Bbb Q/\Bbb Z)\simeq\Bbb Z/n\Bbb Z $ as one can see by examining what can be the image of $\bar 1$ under a map.
On the other hand, if we take as target group a product of infinitely many copies of $\Bbb Q/\Bbb Z$, we sure have infinitely many morphisms.