We present a new X-ray luminosity function of flat-spectrum radio quasars (FSRQs) utilizing the latest {it Swift}/BAT 105-month X-ray source catalog. Contrary to previous studies of FSRQs in the X-ray band, using the luminosity-dependent density evolution model, we find that FSRQs show evolutionary peaks at $zsim1-2$ depending on luminosities. Our result is rather consistent with the evolution of FSRQs seen in the radio and GeV bands, although the number density is a factor of 5--10 smaller. We further explore the contribution of FSRQs to the cosmic MeV gamma-ray background radiation. We find that FSRQs can explain only $sim3$% of the observed MeV gamma-ray background fluxes around 1 MeV, indicating other populations are required. Future MeV gamma-ray observations will be keys for understanding the origin of the MeV gamma-ray background radiation.
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