Of course you could do Cox Regression, but I do not know the details of the study.
Perhaps things observed before treatment started are important as regards
the outcome, and also the length of time before starting treatment may be
important. Suppose, for instance, that during the pre-treatment period there
might be some events of interest (symptoms, whatever): you may want to
include those events in the analysis (as events occurred before treatment)
just as you treat things occurring after starting the treatment. It all
depends on the specifics of your study.
And by the way, if the study did not start with the start of treatment, when
did it begin? With diagnosis? There should have been something causing every
subject to start being observed.
Hector