Several biological systems undergo processes that cannot, as yet, be explained using classical physics. The hypothesis that quantum effects may be responsible for these processes is gaining credibility. In particular, the case of biological hydrogen bonds is particularly interesting. Hydrogen bonds are found in a variety of biological systems, including in the DNA scaffold, in enzymes, and in proteins. Transfer across hydrogen bonds can be modelled as a double-well proton tunnelling problem. However, there is significant concern that quantum coherence cannot survive for biologically relevant timescales in the high-temperature, strongly-interacting cellular environment. We investigate this issue in the framework of open quantum systems by adapting the so-called Caldeira-Leggett model. In open quantum systems, the spectral density describes the response between a system and its environment. By deriving a master equation that is valid for general spectral densities, it is possible to make predictions about the coherence dynamics of a biological hydrogen bond. This will generate insight into the credibility of the hypothesis that quantum mechanics is responsible for certain biological processes.