Apologies in advance if this is a daft question from a healthcare professional with little/no experience of modelling count data over time...

I want to model two counts over time. The start date for each counts is different and in the past (January start, February start, etc). There is a fixed stop data (censorship) for all counts. All data derives from one source and so I guess a single-level model would be ok. I want to estimate the association between the two counts (ie. does one rise as the other does). Is Poisson regression the right method to analyse this time-from-event count data? Is there a function (eg. Cox "in reverse") for time-from-event for count data?

Many thanks for your guidance