Quantum key distribution (QKD) at telecom wavelengths (1260 - 1625 nm) has the potential for fast deployment due to existing optical fibre infrastructure and mature telecom technologies. At these wavelengths, Indium Gallium Arsenide (InGaAs) avalanche photodiode (APD) based detectors are the preferred choice for photon detection. Similar to their Silicon counterparts used at shorter wavelengths, they exhibit fluorescence from recombination of electron-hole pairs generated in the avalanche breakdown process. This fluorescence may open side channels for attacks on QKD systems. Here, we characterize the breakdown fluorescence from two commercial InGaAs single photon counting modules, and find a spectral distribution between 1000 nm and 1600 nm. We also show that by spectral filtering, this side channel can be efficiently suppressed.