A Hubble diagram (HD) has recently been constructed in the redshift range 0<z<6.5 using a non-linear relation between the ultraviolet and X-ray luminosities of QSOs. The Type Ia SN HD has already provided a high-precision test of cosmological models, but the fact that the QSO distribution extends well beyond the supernova range (z<1.8), in principle provides us with an important complementary diagnostic whose significantly greater leverage in z can impose tighter constraints on the distance versus redshift relationship. In this paper, we therefore perform an independent test of nine different cosmological models, among which six are expanding, while three are static. Many of these are disfavoured by other kinds of observations (including the aforementioned Type Ia SNe). We wish to examine whether the QSO HD confirms or rejects these earlier conclusions. We find that four of these models (Einstein-de Sitter, the Milne universe, the Static Universe with simple tired light and the Static universe with plasma tired light) are excluded at the >99% C.L. The Quasi-Steady State Model is excluded at >95% C.L. The remaining four models (Lambda-CDM/wCDM, the R_h=ct Universe, the Friedmann open universe and a Static universe with a linear Hubble law) all pass the test. However, only Lambda-CDM/wCDM and Rh=ct also pass the Alcock-Paczynski (AP) test. The optimized parameters in Lambda-CDM/wCDM are Omega_m=0.20^{+0.24}_{-0.20} and w_{de}=-1.2^{+1.6}_{-infinity} (the dark-energy equation-of-state). Combined with the AP test, these values become Omega_m=0.38^{+0.20}_{-0.19} and w_{de}=-0.28^{+0.52}_{-0.40}. But whereas this optimization of parameters in Lambda-CDM/wCDM creates some tension with their concordance values, the Rh=ct Universe has the advantage of fitting the QSO and AP data without any free parameters.