We construct a model of quintessential inflation in Palatini $R^2$ gravity
employing a scalar field with a simple exponential potential and coupled to
gravity with a running non-minimal coupling. At early times, the field acts as
the inflaton, while later on it becomes the current dark energy. Combining the
scalar sector with an ideal fluid, we study the cosmological evolution of the
model from inflation all the way to dark energy domination. We interpret the
results in the Einstein frame, where a coupling emerges between the fluid and
the field, feeding energy from the former to the latter during the
matter-dominated era. We perform a numerical scan over the parameter space and
find points that align with observations for both the inflationary CMB data and
the late-time behaviour. The final dark energy density emerges from an
interplay between the model parameters, without requiring the extreme
fine-tuning of the cosmological constant in $\Lambda$CDM.
