We conduct a study on the relationship between galaxy environments and their active galactic nuclei (AGN) activity at high redshifts ($2.0<z<4.0$). Specifically, we study the AGN fraction in galaxies residing in a range of environments at these redshifts, from field galaxies to highly overdense peaks in the GOODS-S extragalactic field. Utilizing the extensive photometric and spectroscopic observations in this field, we measure local- and global-overdensities over a large a range of environments, including in several massive (M$_{tot}\geq10^{14.8}$M$_\odot$) protostructures. We employ a multi-wavelength AGN catalog consisting of AGN in nine different categories. Our analysis shows a higher AGN fraction (10.9$^{+3.6}_{-2.3}$%) for galaxies in the highest local-overdensity regions compared to the AGN fraction (1.9$^{+0.4}_{-0.3}$%) of coeval field galaxies (a ~4$\sigma$ difference). This trend of increasing AGN fraction in denser environments relative to the field is present in all redshift bins. We also find this trend consistently present in all five AGN categories that have a sufficient number of AGN to make a meaningful comparison: mid-IR SED, mid-IR color, X-ray luminosity, X-ray-luminosity-to-radio-luminosity-ratio, and optical-spectroscopy. Our results also demonstrate a clear trend of higher (~4x) AGN fractions in denser environments for a given stellar mass. Additionally, we observe the same trend (though at a lower significance) with the global environment of galaxies, measured using a metric based on the projected distance of galaxies from their nearest massive ($M_{tot}>10^{12.8}M_\odot$) overdense ($\sigma_\delta>5.0$) peak, normalized with respect to the size of the peak. These findings indicate that the prevalence of AGN activity is highly dependent on the environment in which a host galaxy resides, even at early times in the formation history of the Universe.