Can the system
$$\cases{\frac{\partial g_1}{\partial x}-\frac{\partial g_2}{\partial y}=f_1\\
\frac{\partial g_2}{\partial x}+\frac{\partial g_1}{\partial y}=f_2}$$
(all the functions are on a region in $\mathbb{R}^2$)
be solved explicitly using the "anti-derivatives" of $f_1$ and $f_2$?
(I need only one solution, and not necessarily the general one)
I'm aware that the system can be transformed to two Poisson equation of the functions separately, but the expression for the solution (as far as I know. maybe I'm wrong) is far from explicit.
Not really, but it depends on what you consider as "explicit". For example, you could take the 2D Fourier transform of your problem and the solution will depend explicitly on the Fourier transforms of $f_1$ and $f_2$.
Nonetheless, I see a more direct trick. I don't know in which context your system of PDEs arises, but it can be seen as inhomogeneous Cauchy-Riemann equations for the complex variable $z = x+iy$ and the function $g(z,\bar{z}) = g_1(x,y) + ig_2(x,y)$, with the source term $f(z,\bar{z}) = \frac{1}{2}(f_1(x,y) + if_2(x,y))$, such that $\partial_{\bar{z}}g = f$, where $\partial_{\bar{z}} = \frac{1}{2}(\partial_x + i\partial_y)$ is the Wirtinger derivative. N.B. : $f$ and $g$ are not holomorphic in that case, that is why they also depend on the conjugate variable $\bar{z}$.
The solution is then given by $$ g(z,\bar{z}) = A(z) + \int_D \frac{f(z',\bar{z}')}{z'-z} \frac{\mathrm{d}z'\wedge\mathrm{d}\bar{z}'}{2\pi i}, $$ where $D\subset\mathbb{R}^2$ is your said region and $A(z)$ an holomorphic function on $D$. So, the antiderivatives of $f_1,f_2$ do not appear in the solution, because it is basically made of the convolution of $f$ with the kernel $\frac{1}{2\pi iz}$.