the curve may start out quite flat for a Poisson process, but the highest rate will always be at zero time lag. It is the lack of a plateau rather than the lack of a peak at several days that may be a deviation from expectations of a Poisson model (i.e., a model with no quake interactions).

You might try making a list of the events at the shortest time lags and closest distances to see whether it is really surprising that these events occur so close in time.