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The growth rate and expansion history of the Universe can be measured from large galaxy redshift surveys using the Alcock-Paczynski effect. We validate the Redshift Space Distortion models used in the final analysis of the Sloan Digital Sky Survey (SDSS) extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 quasar clustering sample, in configuration and Fourier space, using a series of HOD mock catalogues generated using the OuterRim N-body simulation. We test three models on a series of non-blind mocks, in the OuterRim cosmology, and blind mocks, which have been rescaled to new cosmologies, and investigate the effects of redshift smearing and catastrophic redshifts. We find that for the non-blind mocks, the models are able to recover $fσ_8$ to within 3% and $α_\parallel$ and $α_\bot$ to within 1%. The scatter in the measurements is larger for the blind mocks, due to the assumption of an incorrect fiducial cosmology. From this mock challenge, we find that all three models perform well, with similar systematic errors on $fσ_8$, $α_\parallel$ and $α_\bot$ at the level of $σ_{fσ_8}=0.013$, $σ_{α_\parallel}=0.012$ and $σ_{α_\bot}=0.008$. The systematic error on the combined consensus is $σ_{fσ_8}=0.011$, $σ_{α_\parallel}=0.008$ and $σ_{α_\bot}=0.005$, which is used in the final DR16 analysis. For BAO fits in configuration and Fourier space, we take conservative systematic errors of $σ_{α_\parallel}=0.010$ and $σ_{α_\bot}=0.007$.