I was wondering if there was a way to use the API to cause a notification to be triggered on the android device.
Not directly. At least not yet. Right now, these notifications are sent by Google's servers so you can only set up a reminder that will then be sent through the servers, but not generate one locally by the app.
However, we're going to implement offline notifications very soon and also most likely update the public API so this would become an option.
Awesome, thanks for your answer. Will keep an eye out!