The giant elliptical galaxy M87 has been imaged over 30 consecutive days in 2001, 60 consecutive days in 2005-2006, and every 5 days over a 265 day span in 2016-2017 with the Hubble Space Telescope, leading to the detection of 137 classical novae throughout M87. We have identified 2134 globular clusters (GC) in M87 in these images, and carried out searches of the clusters for classical novae erupting in or near them. One GC CN was detected in the 2001 data, while zero novae were found during the 2005-2006 observations. Four candidate GC novae were (barely) detected in visible light during the 2016-2017 observations, but none of the four were seen in near-ultraviolet light, leading us to reject them. Combining these results with our detection of one M87 GC nova out of a total of 137 detected CN, we conclude that such novae may be overabundant relative to the field, but small number statistics dominate this (and all other) searches. A definitive determination of GC CN overabundance (or not) will require much larger samples which LSST should provide in the coming decade.