If it may be forgiven to resurrect a very old question, it's worth pointing out that these are not the "homotopy groups of spheres" that appear in synthetic homotopy theory / homotopy type theory. Those are internal group objects in the $\infty$-topos, not external "ordinary" groups. So in particular it is still true that "$\pi_1(S^1)=\mathbb{Z}$" in the terminal category, because both "$\pi_1(S^1)$" and "$\mathbb{Z}$" denote a (or rather the) object of the terminal category and hence are equal --- it's irrelevant that in that case this internal $\mathbb{Z}$ doesn't have the external $\mathbb{Z}$ as its set of global elements. Similarly, in $\infty Gpd^A$ for a set $A$, we have "$\pi_1(S^1)=\mathbb{Z}$", but the internal object $\mathbb{Z}$ has the external group $\mathbb{Z}^A$ as its set of global elements.

There is a conjecture that internally all the homotopy groups of spheres are always isomorphic to the classical ones. This is still open (indeed the meaning of "always" has yet to be formulated precisely), but it seems to be true for all Grothendieck $\infty$-toposes, because inverse image functors preserve spheres, loop spaces, truncations, and the construction of specific finitely presented abelian groups. Thus, the question about "external" homotopy groups of spheres in $H$, as asked in the question, reduces to asking whether the monad on $Set$ induced by the unique geometric morphism $H\to \infty Gpd$ preserves the corresponding classical homotopy group, and as Charles said this follows in the cohesive case since in that case that monad is the identity.

If it may be forgiven to resurrect a very old question, it's worth pointing out that these are not the "homotopy groups of spheres" that appear in synthetic homotopy theory / homotopy type theory. Those are internal group objects in the $\infty$-topos, not external "ordinary" groups. So in particular it is still true that "$\pi_1(S^1)=\mathbb{Z}$" in the terminal category, because both "$\pi_1(S^1)$" and "$\mathbb{Z}$" denote a (or rather the) object of the terminal category and hence are equal --- it's irrelevant that in that case this internal $\mathbb{Z}$ doesn't have the external $\mathbb{Z}$ as its set of global elements. Similarly, in $\infty Gpd^A$ for a set $A$, we have "$\pi_1(S^1)=\mathbb{Z}$", but the internal object $\mathbb{Z}$ has the external group $\mathbb{Z}^A$ as its set of global elements.

There is a conjecture that internally all the homotopy groups of spheres are always isomorphic to the classical ones. This is still open (indeed the meaning of "always" has yet to be formulated precisely), but it seems to be true for all Grothendieck $\infty$-toposes, because inverse image functors preserve spheres, loop spaces, truncations, and the construction of specific finitely presented abelian groups.