We use a recently proposed luminosity distance measure for relatively nearby active galactic nuclei (AGNs) to test the predicted expansion of the Universe in the $R_{\rm h}=ct$ and $Λ$CDM cosmologies. This comparative study is particularly relevant to the question of whether or not the Universe underwent a transition from decelerated to accelerated expansion, which is believed to have occurred---on the basis of Type Ia SN studies---within the redshift range ($0\lesssim z\lesssim 1.3$) that will eventually be sampled by these objects. We find that the AGN Hubble Diagram constructed from currently available sources does not support the existence of such a transition. While the scatter in the AGN data is still too large for any firm conclusions to be drawn, the results reported here nonetheless somewhat strengthen similar results of comparative analyses using other types of source. We show that the Akaike, Kullback, and Bayes Information Criteria all consistently yield a likelihood of $∼ 84-96\%$ that $R_{\rm h}=ct$ is closer to the ``true" cosmology than $Λ$CDM is, though neither model adequately accounts for the data, suggesting an unnaccounted-for source of scatter.