Hint $S_3$ has a subgroup $H$ with $3$ elements. Since this subgroup $H$ has index $2$ is normal.
Now, both $H$ and $G/H$ have cardinality a prime number, thus are cyclic, but $S_3$ is not abelian.
Note This is not true even for the case $H=Z(G)$. To see this,use the fact that every group with $p$ or $p^2$ elements is abelian (if $p$ is primes), that there exists a non-abelian group with $p^3$ elements, and that the center of a $p$-group is non-trivial.
It follows that $G=$ any non-abelian group with $p^3$ elements, $H=Z(G)$ is also a counterexample.