This isn't currently possible, I'm afraid. The plugin always defaults to the current month. It's on my to-do list to improve this functionality.
However, if you're only using the shortcode, and are ok with "hard-coding" the month the calendar should default to, you could try this extra little plugin I made. It adds
year parameters to the shortcode that allow you to choose the default month.
You'll need to copy / paste the code into Notepad (or similar), save the file as
gce-shortcode-month-year.php, upload it to your
wp-content/plugins folder, then activate it in your WordPress admin.
You'd use the new parameters as follows:
[google-calendar-events type="ajax" month="4" year="2013"]
It's not a great solution, but it might do the trick until I implement this properly.