Take $G=\mathbb{Z}$. Then computing $|\operatorname{Hom}(G, H)|=|H|$ is the same as computing the size of a finitely presented group, and is thus wildly undecidable. This eliminates both the general case you seem to ask about, and the case of fundamental groups of surfaces (replacing $\mathbb{Z}$ with, say $\mathbb{Z}\oplus \mathbb{Z}$ and letting your surface $S$ be a torus).

In other words, this problem seems essentially intractable as you've asked it. On the other hand, if you restrict $H$ to lie in the class of finite groups, then the complexity is bounded above by $$|\text{\# of generators of } G|\cdot |H|\cdot \sum_r t_H(|r|)$$ where the sum is taken over the relations of the given presentation of $G$, and where $t_H(|r|)$ is the time complexity of deciding the word problem in $H$ for a word of length $|r|$. To see this, consider the algorithm which considers all maps $$\{\text{generators of $G$}\to H\}$$ of which there are $$|\text{\# of generators of } G|\cdot |H|,$$ and for each map, checks whether the relations of $G$ are satisfied in $H$. This algorithm has the time complexity described.

So essentially your question is identical to finding the time complexity of solving the word problem in whatever class of groups $H$ belongs to, about which there is tons of literature.

Take $G=\mathbb{Z}$. Then computing $|\operatorname{Hom}(G, H)|=|H|$ is the same as computing the size of a finitely presented group, and is thus wildly undecidable. This eliminates both the general case you seem to ask about, and the case of fundamental groups of surfaces (replacing $\mathbb{Z}$ with, say $\mathbb{Z}\oplus \mathbb{Z}$ and letting your surface $S$ be a torus).

In other words, this problem seems essentially intractable as you've asked it. On the other hand, if you restrict $H$ to lie in the class of finite groups, then the complexity is bounded above by $$|H|^{|\text{\# of generators of } G|}\cdot \sum_r t_H(|r|)$$ where the sum is taken over the relations of the given presentation of $G$, and where $t_H(|r|)$ is the time complexity of deciding the word problem in $H$ for a word of length $|r|$. To see this, consider the algorithm which considers all maps $$\{\text{generators of $G$}\to H\}$$ of which there are $$|H|^{|\text{\# of generators of } G|},$$ and for each map, checks whether the relations of $G$ are satisfied in $H$. This algorithm has the time complexity described.

So essentially your question is identical to finding the time complexity of solving the word problem in whatever class of groups $H$ belongs to, about which there is tons of literature.