In this study, we investigate the cosmological implications of symmetric teleparallel gravity models based on the function f(Q, Lm), where Q is the non-metricity scalar and Lm is the matter Lagrangian. The function f(Q, Lm) allows a natural way of including non-minimal matter-geometry couplings in a second-order formulation. The more minimal and physically motivated polynomial form is analyzed as a possibility to encompass both the early and late-time acceleration of the universe in a single scenario.

In the late universe, we consider the modified Friedmann equations within a spatially flat FLRW background. The model is compared to observational data within the context of a Bayesian analysis. The datasets considered are type Ia supernovae, cosmic chronometers, and BAO observations. The analysis shows that the parameter representing the coupling of matter to geometry is only weakly constrained by the existing observational data, suggesting a near degeneracy to the standard model expansion history.

At high curvature applicable to the early universe, the theory reduces to a quadratic model of non-metricity without extra scalar fields and produces Starobinsky inflation. The predictions for the spectral index (ηs) and tensor-scalar ratio (r) are in accordance with the 2018 Planck constraints.

To extend our study further into the quantum gravity reime, we add corrections due to RGUP in the form of deformation of the metric in a momentum-dependent fashion, which in turn affects matter propagation on top of this same gravitational background, making a non-negligible contribution to the running of some of the inflationary parameters, in particular the spectral index, thus resolving the degeneracy with the classical attractor solution.

In summary, our work demonstrates that the f(Q, Lm) gravity is a well-established set of geometry that is able to account for late-time acceleration, inflation, and quantum geometry in one consistent scheme.