Persistent discrepancies among low and high redshift probes, most notably, the Hubble tension, motivate systematic investigations of cosmologies involving a dynamical dark energy component, rather than a cosmological constant (Lambda) candidature of the latter which gives rise to the standard Lambda-CDM model. Interacting Dark Energy-Matter (IDEM) scenarios provide a more compelling framework in this direction, especially when viewed through modified gravity or their Scalar–Tensor equivalents, however, in the Jordan frame, such interactions do not arise naturally, as the scalar field is non-minimally coupled to gravity. A conformal transformation to the Einstein frame decouples the gravitational and scalar degrees of freedom, but induces a non-minimal coupling between the scalar field and cosmological matter (including the CDM), thereby generating an IDEM-type interaction when the scalar field acts as dark energy. Within this Einstein-frame Scalar–Tensor formulation, we therefore adopt a scenario in which dark energy interacts exclusively with dark matter, whereas the visible matter component remains minimally coupled to both gravity and the IDEM. Thus, the large-scale dilution rates of the dark matter and the visible matter with the cosmological scale factor differ, and this is in principle feasible as we demonstrate explicitly. We carry out a full Bayesian parameter-inference analysis using the standard Metropolis-Hastings algorithm for the Markov Chain Monte Carlo (MCMC) method, by jointly incorporating the Pantheon+ Type~Ia Supernova sample, Observational Hubble Parameter (OHD) measurements via Cosmic Chronometry (CC), and Baryon Acoustic Oscillation (BAO) data from SDSS-IV and DESI-VI. We present updated constraints on the dark-sector coupling parameter and the Hubble constant, and assess the comparative statistics of this Scalar–Tensor IDEM relative to Lambda-CDM using information criteria such as AIC and BIC. The resulting posteriors and reconstructed expansion history illustrate the framework’s impact on the Hubble tension and strengthen its observational viability within Scalar–Tensor gravity.