There are two ways to do this. The easy out-of-the-box way to do this is to use our softphone app which can open a URL upon incoming call. This is documented in our RC for Desktop User Guide (search for "external app"):
http://netstorage.ringcentral.com/guides/rc_for_desktop_user_guide.pdf
If you don't want to use the softphone approach, you can write an app (will need to define it in the Developer Portal first) that listens for incoming calls and opens a URL in the browser. This can be either Browser-Based app, a widget (the RingCentral Salesforce app does this) or a standalone app.