Terrestrial gamma-ray flashes (TGFs) were registered the first time by the NASA's Compton Gamma Ray Observatory. The physical mechanism of TGF generation is not fully known, but there is a consensus among researchers that the radiation is produced by bremsstrahlung of relativistic electrons in the thunderstorm regions of the atmosphere. Therefore, TGFs have been linked to positive-polarity intracloud lightning discharges, strong positive cloud-to-ground discharges or upward discharges from a thundercloud top. The currently operating Fermi Gamma-ray Space Telescope is equipped with a Gamma-ray Burst Monitor that can detect terrestrial gamma-ray flashes. It opens up a new possibility to search for lightning discharges responsible for TGFs. Ground-based lightning monitoring systems in the ELF, LF and VLF ranges can be used for that purpose. The ELF systems are especially useful, since they provide a large monitoring range of several thousand kilometers for strong atmospheric discharges (charge moments above several tens of C km). In this paper we have described the data analysis method for ELF electromagnetic field pulses and applied it to study our first examples of TGFs registered by Fermi GBM coinciding with ELF pulses recorded by the Hylaty ELF station located in the Carpathian Mountains in Poland. Using our ELF electromagnetic wave propagation model we have evaluated charge moments for the two registered events to be 320 and 110 C km and provided upper limits for the remaining events.