The Gaia space telescope allows for unprecedented accuracy for astrometric measurements of stars in the Galaxy. In this work, we explore the sensitivity of Gaia to detect primordial black hole (PBH) dark matter through the glitches that they would create in the apparent trajectory of stars due to astrometric microlensing (AML) effect. By computing the AML event rate, we find regions of the sky that yield the most promising targets for PBH searches with Gaia. We also predict a distribution of observable event durations and centroid shifts of these AML events. We then compute the potential exclusion that could be set on the parameter space of PBHs with a monochromatic mass function. We find that Gaia is most sensitive to PBHs in the vicinity of \(10\,M_\odot\) and has the potential to exclude PBHs that make up as little as \(10^{-3}\) fraction of the dark matter.