Risaliti and Lusso have compiled X-ray and UV flux measurements of 1598 quasars (QSOs) in the redshift range 0.036 ≤ z ≤ 5.1003, part of which, z ∼ 2.4 − 5.1, is largely cosmologically unprobed. In this paper we use these QSO measurements, alone and in conjunction with baryon acoustic oscillation (BAO) and Hubble parameter [H(z)] measurements, to constrain cosmological parameters in six different cosmological models, each with two different Hubble constant priors. In most of these models, given the larger uncertainties, the QSO cosmological parameter constraints are mostly consistent with those from the BAO + H(z) data. A somewhat significant exception is the nonrelativistic matter density parameter Ωm0 where QSO data favor Ωm0 ∼ 0.5 − 0.6 in most models. As a result, in joint analyses of QSO data with H(z) + BAO data the one-dimensional Ωm0 distributions shift slightly toward larger values. A joint analysis of the QSO + BAO + H(z) data is consistent with the current standard model, spatially-flat ΛCDM, but mildly favors closed spatial hypersurfaces and dynamical dark energy. Since the higher Ωm0 values favored by QSO data appear to be associated with the z ∼ 2 − 5 part of these data, and conflict somewhat with strong indications for Ωm0 ∼ 0.3 from most z < 2.5 data as well as from the cosmic microwave background anisotropy data at z ∼ 1100, in most models, the larger QSO data Ωm0 is possibly more indicative of an issue with the z ∼ 2 − 5 QSO data than of an inadequacy of the standard flat ΛCDM model.