Assume that $\mathcal H$ is isomorphic to a Hilbert space $\mathcal G$ by an application $\Phi$, which means that $\Phi$ is an isomorphism (in linear operator sense) between $\mathcal H$ and $\mathcal G$.
Define the inner product on $\mathcal H$ as $$ \left< \Phi(x), \Phi(y) \right>_{\mathcal H} = \left< x, y\right>_{\mathcal G} $$
Can we conclude that $\mathcal H$ is also a Hilbert space?
Yep, this can be called "pulling back" the Hilbert space structure on $G$ to $H$. You can go through and confirm that all the Hilbert space axioms will hold on $H$, but each proof will look the same, in that it will resolve to the fact that the axiom is true on $G$.