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This article focuses on different anisotropic models within the framework of a specific modified $f(\mathcal{R},\mathcal{T},\mathcal{R}_{\zeta\gamma}\mathcal{T}^{\zeta\gamma})$ gravity theory. The study adopts a static spherically symmetric spacetime to determine the field equations for two different modified models: (i) $f(\mathcal{R},\mathcal{T},\mathcal{R}_{\zeta\gamma}\mathcal{T}^{\zeta\gamma})=\mathcal{R}+\eta\mathcal{R}_{\zeta\gamma}\mathcal{T}^{\zeta\gamma}$, and (ii) $f(\mathcal{R},\mathcal{T},\mathcal{R}_{\zeta\gamma}\mathcal{T}^{\zeta\gamma})=\mathcal{R}(1+\eta\mathcal{R}_{\zeta\gamma}\mathcal{T}^{\zeta\gamma})$, where $\eta$ is a constant parameter. To address the additional degrees of freedom in the field equations and obtain their corresponding unique solution, the Durgapal-Fuloria spacetime geometry and MIT bag model are utilized. Matching conditions are applied to determine unknown constants within the chosen spacetime geometry. We adopt a certain range of model parameters to analyze the physical characteristics of the developed models in the interior distribution of a particular compact star candidate 4U 1820-30. Energy conditions and some other tests are also implemented to ensure their viability and stability. Additionally, the disappearing radial pressure constraint is employed to find the values of the model parameter, aligning with the observed information of an array of stars. The study concludes that both of our models are well-behaved and satisfy all necessary conditions, and thus we observe them suitable for the modeling of astrophysical objects.