A key component of the Dark Energy Spectroscopic Instrument (DESI) survey validation (SV) is a detailed visual inspection (VI) of the optical spectroscopic data to quantify key survey metrics. In this paper we present results from VI of the quasar survey using deep coadded SV spectra. We show that the majority (≈70%) of the main-survey targets are spectroscopically confirmed as quasars, with ≈16% galaxies, ≈6% stars, and ≈8% low-quality spectra lacking reliable features. A nonnegligible fraction of the quasars are misidentified by the standard spectroscopic pipeline, but we show that the majority can be recovered using post-pipeline "afterburner" quasar-identification approaches. We combine these "afterburners" with our standard pipeline to create a modified pipeline to increase the overall quasar yield. At the depth of the main DESI survey, both pipelines achieve a good-redshift purity (reliable redshifts measured within 3000 km s-1) of ≈99%; however, the modified pipeline recovers ≈94% of the visually inspected quasars, as compared to ≈86% from the standard pipeline. We demonstrate that both pipelines achieve a median redshift precision and accuracy of ≈100 km s-1 and ≈70 km s-1, respectively. We constructed composite spectra to investigate why some quasars are missed by the standard pipeline and find that they are more host-galaxy dominated (i.e., distant analogs of "Seyfert galaxies") and/or more dust reddened than the standard-pipeline quasars. We also show example spectra to demonstrate the overall diversity of the DESI quasar sample and provide strong-lensing candidates where two targets contribute to a single spectrum.