Based on extensive ab initio calculations and the time-propagation of the nuclear Schrödinger equation, we study the vibrational relaxation dynamics and resulting spectral signatures of the OH stretch vibration of a hydrogen-bonded complex, HCO$^-_2$⋅H$_2$O. Despite their smallness, it has been shown experimentally by Johnson and coworkers that the gas-phase infrared spectra of these types of complexes exhibit much of the complexity commonly observed for hydrogen-bonded systems. That is, the OH stretch band exhibits a significant red shift together with an extreme broadening and a pronounced substructure, which reflects its very strong anharmonicity. Employing an adiabatic separation of time scales between the three intramolecular high-frequency modes of the water molecule and the three most important intermolecular low-frequency modes of the complex, we calculate potential energy surfaces (PESs) of the ground and the first excited states of the high-frequency modes and identify a vibrational conical intersection between the PESs of the OH stretch fundamental and the HOH bend overtone. By performing a time-dependent propagation of the resulting system, we show that the conical intersection affects a coherent population transfer between the two states, the first step of which being ultrafast (60 fs) and irreversible. The subsequent relaxation of vibrational energy into the HOH bend and ground state occurs incoherently but also quite fast (1 ps), although the corresponding PESs are well separated in energy. Owing to the smaller effective mass difference between light and heavy degrees of freedom, the adiabatic ansatz is consequently less significant for vibrations than in the electronic case. Based on the model, we consider several approximations to calculate the measured Ar-tag action spectrum of HCO$^-_2$⋅H$_2$O and achieve semiquantitative agreement with the experiment.