Scaling relations between galactic parameters (e.g., luminosity, mass, metallicity, etc.) represent key pieces of evidence for investigating the processes of galaxy formation and evolution. In most studies, these relations have been obtained for large portions of the galaxies (i.e., on kiloparsec scales), but it is also important to evaluate these relations in smaller scales. In this work, we used optical data cubes of a subsample of nearby galaxies of the DIVING3D survey. These allowed us to analyze the scaling relations involving stellar velocity dispersion, stellar population age, and stellar population metallicity at the nuclear and circumnuclear regions of galaxies (within scales from tens to a few hundreds of parsecs). We detected correlations between the stellar velocity dispersion and the age, metallicity, and total stellar mass. These correlations are independent of galaxy inclinations, considering all morphological types, nuclear activity, and the presence or absence of galactic bars. We detected, for the first time, a correlation between the stellar velocity dispersion and stellar metallicity in the nuclear regions of galaxies. It is found to be qualitatively consistent with the well-known stellar mass-metallicity relation, described in previous studies, on kiloparsec scales. We also noted that barred galaxies tend to show younger and less metal-rich stellar populations than unbarred galaxies in the central regions, which may be a consequence of the bar triggering star formation in the nuclear regions of these objects. However, further studies, with larger samples, are necessary for comparisons between barred and unbarred galaxies, with the same mass and morphological types. Some active galactic nuclei (AGNs) in our sample are positioned above the observed correlation between stellar velocity dispersion and stellar population age, suggesting that their nuclear stellar populations are younger than expected. This may be a consequence of positive AGN feedback, triggering star formation. Conversely, starburst galaxies do not show nuclear stellar populations at ages over one billion years.