An easy way to achieve this is to use a single Document for both text fields.  The Document is the model that stores the data (in this case, plain text).  
Here is a simple example showing two text fields, one in a frame and the other in an option pane.  Update the text in the field in the dialog to see the other text field update instantly.

import java.awt.*;
import java.awt.event.*;
import javax.swing.*;
import javax.swing.border.EmptyBorder;
import javax.swing.text.Document;
public class TwoFieldsUsingOneDocument {
    private JComponent ui = null;
    TwoFieldsUsingOneDocument() {
        initUI();
    }
    public void initUI() {
        if (ui!=null) return;
        ui = new JPanel(new BorderLayout(4,4));
        ui.setBorder(new EmptyBorder(4,4,4,4));
        JTextField frameTextField = new JTextField(15);
        Document doc = frameTextField.getDocument();
        final JTextField dialogTextField = new JTextField(15);
        // get both fields to use a single Document
        dialogTextField.setDocument(doc);
        final JButton showInputDialog = new JButton("Get Input!");
        showInputDialog.setMargin(new Insets(50, 120, 50, 120));
        ActionListener showInputActionListener = new ActionListener() {
            @Override
            public void actionPerformed(ActionEvent e) {
                JOptionPane.showMessageDialog(
                        showInputDialog, 
                        dialogTextField, 
                        "Change text", 
                        JOptionPane.QUESTION_MESSAGE);
            }
        };
        showInputDialog.addActionListener(showInputActionListener);
        ui.add(showInputDialog);
        ui.add(frameTextField, BorderLayout.PAGE_END);
    }
    public JComponent getUI() {
        return ui;
    }
    public static void main(String[] args) {
        Runnable r = new Runnable() {
            @Override
            public void run() {
                try {
                    UIManager.setLookAndFeel(UIManager.getSystemLookAndFeelClassName());
                } catch (Exception useDefault) {
                }
                TwoFieldsUsingOneDocument o = new TwoFieldsUsingOneDocument();
                JFrame f = new JFrame(o.getClass().getSimpleName());
                f.setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
                f.setLocationByPlatform(true);
                f.setContentPane(o.getUI());
                f.pack();
                f.setMinimumSize(f.getSize());
                f.setVisible(true);
            }
        };
        SwingUtilities.invokeLater(r);
    }
}