Theoretical estimates are given along with the first experimental results on the observation of resonance fluorescence in nuclei of rubidium isomer under conditions of laser plasma X-ray pumping of the contiguous transition with an energy of 3.4 keV. The laser plasma is prepared by irradiating a silver target by a powerful radiation of a Nd laser with a pulse duration of 600 ps. It is demonstrated how one can use the recorded number of emitted γ quanta to determine the probability of low-energy nuclear transition excited by laser plasma X rays.