Yes you can (if you have a gmail account as then it will sync all your gmail stuff (calender, maps, etc))
If you go into settings, then "Accounts & sync", then under your gmail account, there should be a "Sync Calendar" which is already ticked.
You can check here to see if it's been synced: https://www.google.com/calendar
Strange, do you have a Google Calendar app on your phone?
And definitely nothing here?
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Link to google calendar app if you don't have: https://play.google.com/store/apps/...yLDEsImNvbS5nb29nbGUuYW5kcm9pZC5jYWxlbmRhciJd