In the close vicinity of a hot solid, at distances smaller than the thermal wavelength, a strong electromagnetic energy density exists because of the presence of evanescent field. Here we explore the possibility to harvest this energy using graphene-based pyroelectric conversion devices made with an active layer encapsulated between two graphene field-effect transistors (GFETs) deposited on the source and on the cold sink. By tuning the bias voltage applied to the gates of these transistors, the thermal state and the spontaneous polarization of the active layer can be controlled at kHz frequencies. We demonstrate that the power density generated by these conversion systems can reach 1300 W/m^2 using pyroelectric Ericsson cycles, a value which surpasses the current production capacity of near-field thermophotovoltaic conversion devices by more than five orders of magnitude with low grade heat sources (T<500 K) and small temperature differences (DT~100 K).