Exploring galaxy evolution in dense environments, such as proto-clusters, is pivotal for understanding the mechanisms that drive star formation and the quenching of star formation. This study provides insights into how two over-densities could have impacted the physical properties, such as the star formation rate, stellar mass, morphology, and the evolution of their members, particularly members characterised by a quenching of star formation. We focus on the over-densities identified at $3<z<4$ in the Chandra Deep Field South (CDFS) and in the Ultra Deep Survey (UDS) regions of the VIMOS (VIsible MultiObject Spectrograph) Ultra Deep Survey (VANDELS). Our methodology involves the analysis of the spectral energy distribution of the members of the over-densities and of the galaxies in the field. We relied on Bayesian analysis techniques BEAGLE and BAGPIPES to study the best-fit physical parameters and the rest-frame U-V and V-J colours (UVJ). This approach allowed us to separate quenched and star-forming galaxies based on the UVJ diagram and by estimating their specific star formation rate (sSFR). We used the TNG300 simulation to interpret our results. We find that two out of 13 over-densities host quenched galaxies, with red rest-frame U-V colour and low sSFR. The physical properties of them are consistent with those of massive passive galaxies from the literature. The quenched members are redder, older, more massive, and show a more compact morphology than the other galaxy members. The two over-densities, with the highest-density peaks at $z and $z respectively, have dark matter halo masses consistent with being proto-clusters at $z and they each host an active galactic nucleus (AGN). We found five AGNs in the structure at $z and three AGNs in the one at $z In comparison to quenched galaxies in the field, our massive quenched members show a higher local density environment. By using the IllustrisTNG simulation (TNG300), we find that proto-cluster structures with quenched galaxies at high redshift are likely to evolve into a structure with a higher fraction of passive galaxies by $z = The two over-densities studied here host massive quenched galaxies in their highest-density peaks and AGNs. By following the evolution of the passive galaxies in the simulated proto-clusters at $z = 3$ from the TNG300 simulation, we find that the median of their sSFRs was larger than 10$^ $ yr$^ $ at $z=6$ and the median mass growth rate was 96<!PCT!> from $z=6$ to $z=3$. In 20<!PCT!> of the simulated proto-clusters, the passive galaxy had already accreted 10-20<!PCT!> of the mass at $z=6$, with SFRs $>100$ M$_ odot $ yr$^ $ at $z=8$. The conditions for this favorable mass assembly could be the galaxy interactions and the high gas accretion rate in the dense environment. As a consequence, the quenching of the star formation at $z=3$ could be driven by the black hole mass growth and AGN feedback. This scenario is consistent with the properties of the two quenched galaxies we find in our two over-densities at $z sim3$.
