Under the account settings in the fitbit app, make sure that automatic time zone is checked. If it is, then any time issues you are having are probably related to the phone and not the fitbit app. As long as the fitbit is synced to the phone, the time will adjust accordingly.
The fitbit app can't alter your phone's time so if the phone is having issues with the time, then the problem will need a fix that is a little more in depth than just an app settings change and you will probably need to contact your phone's support desk or your mobile network provider's support desk.
