According to Kuratowski's theorem, all non planar graphs must contain a subgraph that can be obtained from $K_{3,3}$ or $K_5$ (where $K_5$ represents complete graph containing $5$ vertices and $K_{3,3}$ is a $3$, $3$ complete bipartite graph). Does $Q_4$ contain a subgraph of $K_{3,3}$ or $K_5$?
If yes, then where is it located?
In the graph $Q_n$ you can easily find an subdivision of the graph $K_{n+1}$. Consider the vertices $v_0=0\dots0$ and $v_i=\overline{0\dots010\dots0}$ for $i\in[1,n]$ with the edges $v_0v_i$ and the subdivision $v_iv_j$ through the vertex $\overline{0\dots0\underbrace{1}_i0\dots0\underbrace{1}_j0\dots0}$.