The "correct" way to do this is to use the native Activity framework, but Processing itself is based on an activity, so it is not possible to do that.

As the wiki says, you have to override the back button like so:

1. void keyPressed() {
2.   // doing other things here, and then:
3.   if (key == CODED && keyCode == KeyEvent.KEYCODE_BACK) {
4.     keyCode = 0;  // don't quit by default
5.   }
6. }

However, it appears that even that doesn't work.
In order to get this to work (I'm pretty sure) you have to export the application as a signed application that is not in debug mode, and you have to disable (something like) Settings > Applications > Development > Long Press Back Button Quit on the Android Device.

see,

https://forum.processing.org/topic/back-button-in-android

the code from David Wilson (the last answer) works well for me.