Let $X$ be a non-compact connected real surface (2 dimensional real manifold), then we have the following theorem
Theorem (Johansson 1931)
If $X$ is as above then $\pi_1(X)$ is free. In particular there exists an isomorphism $\pi_1(X)\cong\ast_{i\in I}\mathbb{Z}\cdot i$, with $I$ possibly infinite.
Recall also the following fact from Algebraic Topology:
Theorem (Hurewicz's Theorem)
Let $X$ be a path-connected topological space, then there exists an isomorphism of abelian groups \begin{equation*}\Phi:H_1(X,\mathbb{Z})\to\pi_1(X)^{\text{ab}} \end{equation*}where $\pi_1(X):=\pi_1(X)/[\pi_1(X),\pi_1(X)]$
Assuming this two theorems I would like to prove the following fact:
Proposition
Let $X$ be a connected Riemann surface, then \begin{equation*}\pi_1(X)\cong\{0\}\iff H_1(X,\mathbb{Z})\cong\{0\}\end{equation*}
Proof: The only non trivial implication is when $X$ is homologically trivial, so assume $H_1(X,\mathbb{Z})\cong\{0\}$
If $X$ is compact, then $X$ is homeomorphic to a closed orientable connected real surface $\Sigma_g$ of genus $g\ge 0$. In particular we have $\{0\}\cong H_1(X,\mathbb{Z})\cong\mathbb{Z}^{\oplus^{2g}}$ and this is possible iff $g=0$ which implies that $X$ is homeomorphic to $S^2$ and so it is simply-connected.
If $X$ is non-compact, then by the Johansson's Theorem we have $\pi_1(X)\cong\ast_{i\in I}\mathbb{Z}\cdot i$. Moreover, by using Hurewicz's Theorem (taking $X:=\bigvee_{i\in I}S^1$) one can prove that $\bigg(\ast_{i\in I}\mathbb{Z}\cdot i\bigg)^{\text{ab}}\cong\bigoplus_{i\in I}\mathbb{Z}\cdot i$. So we have \begin{equation*}0\cong H_1(X,\mathbb{Z})\cong\pi_1(X)^{\text{ab}}=\bigg(\ast_{i\in I}\mathbb{Z}\cdot i\bigg)^{\text{ab}}\cong\bigoplus_{i\in I}\mathbb{Z}\cdot i \end{equation*}but this is possible iff $I=\emptyset$ which implies $\pi_1(X)\cong\{0\}$
Is my proof correct? Of course, this proof relies on dimensional constraints and it is no longer true in real dimension 3 (Poincarè Homology 3-sphere). In particular this implies that if $G$ is a perfect group isomorphic to the fundamental group of a connected real surface, then it is trivial.