First note that $S^3 = \{(x,y,-y-x)\mid |x|^2+|y|^2 + |x+y|^2 = 1\}$. Since the action of $G=\mathbb{Z}_3$ permutes the coordinates, it's clear that $G$ preserves $S^3$. Now, suppose $e\neq g\in G$ and that $g(x,y,-y-x) = (x,y,-y-x)$. Then in particular, we also have $g^2(x,y,-y-x) = (x,y,-y-x)$. It follows that we must have $x = y = -y-x$ and from this it follows that the fixed point was $(0,0,0)$, which is not an element of $S^3$.
Note that this proof doesn't use the fact that we're looking at the unit sphere, only that the sphere has nonzero radius.
Next, notice that $H \cong C(S^3) = S^3\times[0,\infty)/$~ where ~ collapses all of $S^3\times\{0\}$ to a point. The map establishing this homemorphism can be defined as follows: Every non $0$ point $q$ in $H$ determines a unique ray emanating from $0$. This ray will pierce the sphere $S^3$ in precisely one point $f(q)$. Now, map $H$ to $C(S^3)$ by sending $q$ to $(f(q), |q|^2)$ if $q\neq 0$ and sending $0$ to $S^3\times\{0\}$. I leave it to you to prove this is a homeomorphism.
Further, this homemorphism is $G$ equivariant if $G$ acts on $C(S^3)$ by simply copying the $G$ action on each $S^3$. That is, $g(p, t) = (gp, t)$. (Again, I leave this to you to prove). This should easily allow you to construct a homeomorphism from $H/G \cong C(S^3)/G$ to $C(S^3/G)$.