I don't know if it is your case, but if you're using Team Calendar you can sync with Google Calendar using this: https://confluence.atlassian.com/display/TEAMCAL/Subscribing+to+Google+Calendars+From+Confluence.
Team Calendar is an add-on for Confluence (so you will need to also have Confluence installed), but you may add JIRA issue dates to it, as you can see here: https://confluence.atlassian.com/display/TEAMCAL/Adding+JIRA+events+to+a+Team+Calendar#AddingJIRAeventstoaTeamCalendar-AddingJIRAIssueDates
If you want to know more about Team Calendar, you can check it here: https://www.atlassian.com/software/confluence/team-calendars.
I hope it was helpful.
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