The answer to this is going to depend on exactly what you're trying to do.
Are you trying to do this from Java mode? If so, google something like "java launch webpage".
Are you trying to do this from JavaScript mode, or with Processing.js or p5.js? If so, google something like "JavaScript fullscreen" and you'll find examples like this and this.
That will allow you to open a webapge in fullscreen mode. Then going back to your Processing sketch again depends on whether you're using Java or JavaScript.
If you're using Java, this is going to be pretty difficult, as you aren't going to be able to get events from the webpage to your Java code. You could try using an applet, but I wouldn't really recommend that since applets are pretty much dead anyway.
If you're using JavaScript, then it's a little easier. You can simply detect the button click in the webpage (by adding an onclick callback) and then call whatever functions you want to close the webpage and reactivate your Processing sketch.
It's hard to answer general "how do I do this" type questions. Stack Overflow is more designed for specific "I tried X, expected Y, but got Z instead" type questions. I really recommend that you just try something out and post an MCVE showing exactly where you're stuck. Good luck.