It is well-known that collective electronic excitations in fullerene C60 are manifested as Mie plasmons, and in graphene (the limiting case of an infinitely large fullerene), the collective excitations are of the plasmon-polariton type. How the properties of plasmons change in fullerenes with intermediate sizes is poorly understood. This problem is considered in the current paper in the framework of the GW approximation on the example of fullerenes C60, C240, and C540. The calculations predict that a high-frequency plasmon resonance begins to form in C240, and in C540, the intensity of this resonance becomes comparable to the intensity of Mie plasmon resonance. We associate this resonance with the incipient plasmon-polariton oscillations. The paper is the first identification and study of plasmon-polaritons in the excitation spectrum of fullerenes.