I went for a run this morning at 6am. Garmin Connect now tells me I actually went at 5am. And apparently ran an hour earlier for all the runs I have taken over the last few months. My watch gives the right time.
What device are you using? You have your time zone set correctly within Garmin Connect as well as your device? The device should be getting this automatically unless you've changed it to "manual."
Same problem, except when I go running at 10am, connect claims it was 11 am. When I edit the time to be correct, it shows right in the listing, but not when I open the activity.
This has been happening for I think maybe 3 weeks. I think at the same time there was some update on connect mobile.
Time-zone is set correct on connect, and phone too. In device it is set to automatic.