The inclusive production of pseudoscalar heavy quarkonia (ηc, ηb, and Bc) via photon-photon fusion in heavy ion ultraperipheral collision (UPC) are calculated to QCD next-to-leading order in the framework of nonrelativistic QCD (NRQCD). The total cross section of ηc produced in Pb-Pb UPC is 194 nb and 1275 nb at nucleon-nucleon c.m. energies SNN=5.52 TeV and 39.4 TeV, respectively. The cross sections for ηb and Bc mesons are more than two to three orders of magnitude smaller. We make a detailed phenomenological analysis on the ηc production; the uncertainties caused by the renormalization scale and the charm quark mass, the cross sections in other ultraperipheral nucleon-nucleon colliding systems, and the transverse momentum distribution are discussed. At the coming HL-LHC and future FCC, the heavy ion UPC opens another door of the study on the production of heavy quarkonium. Published by the American Physical Society 2024