A passively Q-switched laser produces regular pulses above threshold. Here we show that this type of laser, described by the Yamada model, exhibits excitability below threshold. There is a stable equilibrium, the off-solution, from which the system can be triggered by a sufficiently large, but small perturbation, to produce a single pulse after which it settles back to the off-solution. In order to study possible applications, such as pulse reshaping and clock recovery, we consider the reaction of the laser to a triggering input pulse. Furthermore, we demonstrate that when the laser is subjected to injected optical noise below threshold it reacts by producing a train of pulses, whose coherence is maximal and, equivalently, the normalized jitter is minimal for a particular noise level. This shows that the Yamada model with optically injected noise constitutes an example of all-optical coherence resonance.