We report the discovery of 11 newly found quasars behind the stellar disks of the spiral galaxies M31 and M33 in the fields covered by the Local Group Galaxy Survey. Their redshifts range from 0.37 to 2.15. Most are X-ray, ultraviolet, and infrared sources. We also report the discovery of five normal background galaxies. Most of these objects were observed owing to their anomalous colors, as part of a program (reported elsewhere) to confirm spectroscopically candidate red supergiant plus B-star binaries; others were discovered as part of our identification of early-type massive stars based upon their optical colors. There are 15 previously known quasars in the same fields, for a grand total of 26, 15 behind M31 and 11 behind M33. Of these, only eight were discovered as part of surveys for quasars; the rest were found accidentally. The quasars are well distributed in the M31 and M33 fields, except for the inner regions, and have the potential for being good probes of the interstellar medium in these stellar disks, as well as serving as zero-point calibrators for Gaia parallaxes.