  Many-body perturbation theory methods, such as the $G_0W_0$ approximation,
are able to accurately predict quasiparticle (QP) properties of several classes
of materials. However, the calculation of the QP band structure of
two-dimensional (2D) semiconductors is known to require a very dense BZ
sampling, due to the sharp $q$-dependence of the dielectric matrix in the
long-wavelength limit ($\mathbf{q} \to 0$). In this work, we show how the
convergence of the QP corrections of 2D semiconductors with respect to the BZ
sampling can be drastically improved, by combining a Monte Carlo integration
with an interpolation scheme able to represent the screened potential between
the calculated grid points. The method has been validated by computing the band
gap of three different prototype monolayer materials: a transition metal
dichalcogenide (MoS$_2$), a wide band gap insulator (hBN) and an anisotropic
semiconductor (phosphorene). The proposed scheme shows that the convergence of
the gap for these three materials up to 50meV is achieved by using $\k$-point
grids comparable to those needed by DFT calculations, while keeping the grid
uniform.
