Yes, In-app billing API version 3 supports a trial period! That I couldn't find whether there is an API call to give the kind of details on the trial period you need in the documentation doesn't mean there isn't such a class/method.
However, relying only on the remote Google service for the subscription status means the user will not be notified when not connected to a network. Maybe lack of notification when not on a network is not a problem. But, knowing of this limitation, I would record and keep the start date locally, as well.
Once you have captured the results of the API call that allows the user to start the trial period, you can save that data locally. There are three 'easy' ways to keep local data: via preferences (easiest), write a local file (a little harder), or setup and populate an sqlite db (yet, even harder). I use preferences all the time and is quite easy.
If you want to make the 'local' data accessible to multiple Android devices (if your subscription scheme supports multiple devices), you will need to PUT this data to a remote server, instead. And, GET the data at activity start.
Keeping it simple - data on a local device - read the local preference to see when the trial started, calculate the number of days remaining, and present a message. Also use the Google API call to confirm the subscription status. In case of a conflict, update the local preference, accordingly.
With this strategy, the user can be notified at application/activity launch on the trial period time remaining without using a Google API to get details of the trial period; and as a bonus, regardless of whether the app user is on a network, or not.