Context. The existence of axion quark nuggets is a potential consequence of the axion field, which provides a possible solution to the charge-conjugation parity violation in quantum chromodynamics. In addition to explaining the cosmological discrepancy of matter-antimatter asymmetry and a visible-to-dark-matter ratio of Ω_dark/Ω_visible ≃ 5, these composite compact objects are expected to represent a potentially ubiquitous electromagnetic background radiation by interacting with ordinary baryonic matter. We conducted an in-depth analysis of axion quark nugget-baryonic matter interactions in the environment of the intracluster medium in the constrained cosmological Simulation of the LOcal Web (SLOW). Aims. Here, we aim to provide upper limit predictions on electromagnetic counterparts of axion quark nuggets in the environment of galaxy clusters by inferring their thermal and non-thermal emission spectrum originating from axion quark nugget-cluster gas interactions. Methods. We analyzed the emission of axion quark nuggets in a large sample of 161 simulated galaxy clusters using the SLOW simulation. These clusters are divided into a sub-sample of 150 galaxy clusters, ordered in five mass bins ranging from 0.8 to 31.7 × 10¹⁴ M_⊙, along with 11 cross-identified galaxy clusters from observations. We investigated dark matter-baryonic matter interactions in galaxy clusters in their present stage at the redshift of z = 0 by assuming all dark matter consists of axion quark nuggets. The resulting electromagnetic signatures were compared to thermal Bremsstrahlung and non-thermal cosmic ray (CR) synchrotron emission in each galaxy cluster. We further investigated individual frequency bands imitating the observable range of the WMAP, Planck, Euclid, and XRISM telescopes for the most promising cross-identified galaxy clusters hosting detectable signatures of axion quark nugget emission. Results. We observed a positive excess in the low- and high-energy frequency windows, where thermal and non-thermal axion quark nugget emission can significantly contribute to (or even outshine) the emission of the intracluster medium (ICM) in frequencies up to ν_T ≲ 3842.19 GHz and ν_T ϵ [3.97, 10.99] × 10¹⁰GHz, respectively. Emission signatures of axion quark nuggets are found to be observable if CR synchrotron emission of individual clusters is sufficiently low. The degeneracy in the parameters contributing to an emission excess makes it challenging to offer predictions with respect to pinpointing specific regions of a positive axion quark nugget excess; however, a general increase in the total galaxy cluster emission is expected based on this dark matter model. Axion quark nuggets constitute an increment of 4.80% of the total galaxy cluster emission in the low-energy regime of ν_T ≲ 3842.19 GHz for a selection of cross-identified galaxy clusters. We propose that the Fornax and Virgo clusters represent the most promising candidates in the search for axion quark nugget emission signatures. Conclusions. The results from our simulations point towards the possibility of detecting an axion quark nugget excess in galaxy clusters in observations if their signatures can be sufficiently disentangled from the ICM radiation. While this model proposes a promising explanation for the composition of dark matter, with the potential to have this outcome verified by observations, we propose further changes that are aimed at refining our methods. Our ultimate goal is to identify the extracted electromagnetic counterparts of axion quark nuggets with even greater precision in the near future.