We construct a cosmological model from the inception of the Friedmann-Lem^aitre-Robertson-Walker metric into the field equations of the $f(R,L_m)$ gravity theory, with $R$ being the Ricci scalar and $L_m$ being the matter lagrangian density. The formalism is developed for a particular $f(R,L_m)$ function, namely $R/16pi +(1+sigma R)L_{m}$, with $sigma$ being a constant that carries the geometry-matter coupling. Our solutions are remarkably capable of evading the Big-Bang singularity as well as predict the cosmic acceleration with no need for the cosmological constant, but simply as a consequence of the geometry-matter coupling terms in the Friedmann-like equations.
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