After the identification of the gamma-ray blazar TXS 0506+056 as the first compelling IceCube neutrino source candidate, we perform a systematic analysis of all high-energy neutrino events satisfying the IceCube realtime trigger criteria. We find one additional known gamma-ray source, the blazar GB6 J1040+0617, in spatial coincidence with a neutrino in this sample. The chance probability of this coincidence is 30% after trial correction. For the first time, we present a systematic study of the gamma-ray flux, spectral and optical variability, and multi-wavelength behavior of GB6 J1040+0617 and compare it to TXS 0506+056. We find that TXS 0506+056 shows strong flux variability in the Fermi-LAT gamma-ray band, being in an active state around the arrival of IceCube-170922A, but in a low state during the archival IceCube neutrino flare in 2014/15. In both cases the spectral shape is statistically compatible ($leq 2sigma$) with the average spectrum showing no indication of a significant relative increase of a high-energy component. While the association of GB6 J1040+0617 with the neutrino is consistent with background expectations, the source appears to be a plausible neutrino source candidate based on its energetics and multi-wavelength features, namely a bright optical flare and modestly increased gamma-ray activity. Finding one or two neutrinos originating from gamma-ray blazars in the given sample of high-energy neutrinos is consistent with previously derived limits of neutrino emission from gamma-ray blazars, indicating the sources of the majority of cosmic high-energy neutrinos remain unknown.
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