Recently, atomically well-defined cove-shaped graphene nanoribbons have been obtained using bottom-up synthesis. These nanoribbons have an optical gap in the visible range of the spectrum which make them candidates for donor materials in photovoltaic devices. From the atomistic point of view, their electronic and optical properties are not clearly understood. Therefore, in this work we carry out ab-initiodensity functional theory calculations combine with many-body perturbation formalism to study their electronic and optical properties. Through the comparison with experimental measurements, we show that an accurate description of the nanoribbon's optical properties requires the inclusion of electron-hole correlation effects. The energy, binding energy and the corresponding excitonic transitions involved are analyzed. We found that in contrast to zigzag graphene nanoribbons, the excitonic peaks in the absorption spectrum are a consequence of a group of transitions involving the first and second conduction and valence bands. Finally, we estimate some relevant optical properties that strengthen the potential of these nanoribbons for acting as a donor materials in photovoltaic.