There are many possible approaches, I list below a couple of the ways that you could go about this:
i) You could try and apply a RegARIMA model. You're probably going to have to have 2 or 3 covariates, as it appears this 'spike' doesn't occur on the same month each year.
Better yet, if you had some other variable that was correlated with the occurrence of the spike, you would use this in your regARIMA model. Indeed, looking at your data, I wouldn't be surprised if the occurrence of the spike was correlated with Easter. It just so happens this is exactly what you observe in many time series, as Easter occurs in March or April. And, in your data, the spike appears in the third or fourth month.
For more information on this approach, you'd be best served to look how it is implemented in X12ARIMA by the US Census Bureau (or any other NSO).
ii) you could give up the idea that the data is not seasonal. If you are getting a spike at the same time each year, that would seem to indicate seasonality... have you done a test for (moving) seasonality? If the data is seasonal, you of course would apply a SARIMA model.