The definition of a generating set $S$ of a subgroup $H\leq G$ is such that if $S$ is a subset of $H$, then for any subgroup $K\le G$ containing $S$, $H\leq K$. Now, for $\mathcal{Q}_8$, if $S=\{i, j\}$ then $\mathcal{Q}_8$ is not a proper subgroup of any group. So I'm not sure how to build this proof based on the definition.
I would appreciate some hints.