A simple way to achieve a sidebar upcoming events for your google calendar is to use the embeddable option – click on the image below for options used. It uses the eccentric dates calendar as an example.
I suggest not using it on a sidebar that appears on every page – only front page, or only use on a single page. I don’t whether the browser would do adequate cacheing (as my plugin does) or whether you would be making a call to google each time.
You’d need to either update the theme sidebar, or use a widget that allows html code, or put the code in a single page.