When you click on the mute button you make that window active (i.e. the focus for all window events) to give the textfield focus then you need to make the window that contains it active.

Since GWindow inherits from java.awt.Frame which in turn inherits from java.awt.Window then methods in these classes are available for us andafter a bit of looking I found the toFront() method which appears to do the job - try the code below

When you click on the mute button then the textfield gets focus. The setFocus will highlight all the text so a keypress will replace it. If you want to append then press the right arrow key.
 

1. // Need G4P library
2. import g4p_controls.*;
3. 
   
4. public void setup(){
5.   size(480, 320, JAVA2D);
6.   createGUI();
7.   customGUI();
8.   // Place your setup code here
9.  
10. }
11. 
    
12. public void draw(){
13.   background(230);
14.  
15. }
16. // Use this method to add additional statements
17. // to customise the GUI controls
18. public void customGUI(){
19. }
20. 
    
21.  void myBtnEvents(GButton button) { //_CODE_:button1:12356:
22.      // It is safe to enter your event code here 
23.  } //_CODE_:button1:12356:
24. 
    
25. public void button1_click1(GButton source, GEvent event) { //_CODE_:button1:872060:
26.   window1.toFront();
27.   textfield1.setFocus(true);
28. } //_CODE_:button1:872060:
29. 
    
30. synchronized public void win_draw1(GWinApplet appc, GWinData data) { //_CODE_:window1:493117:
31.   appc.background(230);
32. } //_CODE_:window1:493117:
33. 
    
34. // Create all the GUI controls.
35. // autogenerated do not edit
36. public void createGUI(){
37.   G4P.messagesEnabled(false);
38.   G4P.setGlobalColorScheme(GCScheme.BLUE_SCHEME);
39.   G4P.setCursor(ARROW);
40.   if(frame != null)
41.     frame.setTitle("Sketch Window");
42.   button1 = new GButton(this, 66, 41, 80, 30);
43.   button1.setText("Face text");
44.   button1.addEventHandler(this, "button1_click1");
45.   window1 = new GWindow(this, "Window title", 0, 0, 240, 120, false, JAVA2D);
46.   window1.addDrawHandler(this, "win_draw1");
47.   textfield1 = new GTextField(window1.papplet, 40, 45, 160, 30, G4P.SCROLLBARS_NONE);
48.   textfield1.setOpaque(true);
49. }
50. 
    
51. // Variable declarations
52. // autogenerated do not edit
53. GButton button1;
54. GWindow window1;
55. GTextField textfield1;