How to add more panels to a java gridlayout

You simply have to convert it from a form to a panel, as follows:

package com.stackoverflow.es.question371169;

import java.awt.Container;
import java.awt.Dimension;
import java.awt.GridLayout;
import java.awt.Toolkit;
import javax.swing.ImageIcon;
import javax.swing.JButton;

public class Tablero extends javax.swing.JPanel {

    private Container contenedor;
    private GridLayout layout;
    private JButton[][] botones;

    public Tablero() {
        initComponents();
        layout = new GridLayout(8, 8);
        contenedor = this;
        contenedor.setLayout(layout);
        botones = new JButton[8][8];

        Dimension pantalla = Toolkit.getDefaultToolkit().getScreenSize();
        int height = pantalla.height;
        int width = pantalla.width;
        setSize(693, 557);

        //setLocationRelativeTo(null);

        ImageIcon icono = null;
        java.net.URL imgURL1 = Tablero.class.getResource("/imagenes/reina.jpg");

        if (imgURL1 != null) {
            icono = new ImageIcon(imgURL1); //Usa imgURL1 o imgURL2
        } else {
            System.out.println("No se pudo cargar la imagen.");
        }

        for (int i = 0; i < 8; i++) {
            for (int j = 0; j < 8; j++) {

                if (i == 7) {
                    botones[i][j] = new JButton(icono);
                } else {
                    if (i == 0) {
                        botones[i][j] = new JButton("0");

                    } else {
                        botones[i][j] = new JButton("0");
                    }

                }

                contenedor.add(botones[i][j]);
            }
        }

        // contenedor.add(new JButton("siguiente"));
//      this.jPanel1.setBounds(0, 0, 512, 512);
//      this.jPanel1.setBackground(new java.awt.Color(190, 190, 190));
    }
//  private void jPanel1MouseClicked(java.awt.event.MouseEvent evt) {                                    
//      Dibujar t = new Dibujar(jPanel1.getGraphics(), this.jPanel1.getWidth());
//   }

    /**
     * This method is called from within the constructor to initialize the form.
     * WARNING: Do NOT modify this code. The content of this method is always
     * regenerated by the Form Editor.
     */
    @SuppressWarnings("unchecked")
    // <editor-fold defaultstate="collapsed" desc="Generated Code">                          
    private void initComponents() {

        

        javax.swing.GroupLayout layout = new javax.swing.GroupLayout(this);
        setLayout(layout);
        layout.setHorizontalGroup(
                layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
                        .addGap(0, 693, Short.MAX_VALUE)
        );
        layout.setVerticalGroup(
                layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
                        .addGap(0, 557, Short.MAX_VALUE)
        );

        
    }// </editor-fold>                        

    /**
     * @param args the command line arguments
     */
    

    // Variables declaration - do not modify                     
    private javax.swing.JPanel jPanel36;
    private javax.swing.JPanel jPanel37;
    // End of variables declaration                   
}

As you may have noticed, you remove all references to things that are from JFrame

From now on you can without worrying about embedding it in your main frame, for example:

package com.stackoverflow.es.question371169;

public class ElAjedrez extends javax.swing.JFrame {

    /**
     * Creates new form ElAjedrez
     */
    public ElAjedrez() {
        initComponents();
    }

    /**
     * This method is called from within the constructor to initialize the form.
     * WARNING: Do NOT modify this code. The content of this method is always
     * regenerated by the Form Editor.
     */
    @SuppressWarnings("unchecked")
    // <editor-fold defaultstate="collapsed" desc="Generated Code">                          
    private void initComponents() {

        tablero1 = new com.stackoverflow.es.question371169.Tablero();
        jPanel1 = new javax.swing.JPanel();
        jButton1 = new javax.swing.JButton();
        jButton2 = new javax.swing.JButton();

        setDefaultCloseOperation(javax.swing.WindowConstants.EXIT_ON_CLOSE);
        getContentPane().add(tablero1, java.awt.BorderLayout.CENTER);

        jButton1.setText("jButton1");

        jButton2.setText("jButton2");

        javax.swing.GroupLayout jPanel1Layout = new javax.swing.GroupLayout(jPanel1);
        jPanel1.setLayout(jPanel1Layout);
        jPanel1Layout.setHorizontalGroup(
            jPanel1Layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
            .addGroup(jPanel1Layout.createSequentialGroup()
                .addContainerGap()
                .addComponent(jButton1)
                .addPreferredGap(javax.swing.LayoutStyle.ComponentPlacement.RELATED, 231, Short.MAX_VALUE)
                .addComponent(jButton2)
                .addContainerGap())
        );
        jPanel1Layout.setVerticalGroup(
            jPanel1Layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
            .addGroup(jPanel1Layout.createSequentialGroup()
                .addContainerGap()
                .addGroup(jPanel1Layout.createParallelGroup(javax.swing.GroupLayout.Alignment.BASELINE)
                    .addComponent(jButton1)
                    .addComponent(jButton2))
                .addContainerGap(64, Short.MAX_VALUE))
        );

        getContentPane().add(jPanel1, java.awt.BorderLayout.SOUTH);

        pack();
    }// </editor-fold>                        

    /**
     * @param args the command line arguments
     */
    public static void main(String args[]) {
        /* Set the Nimbus look and feel */
        //<editor-fold defaultstate="collapsed" desc=" Look and feel setting code (optional) ">
        /* If Nimbus (introduced in Java SE 6) is not available, stay with the default look and feel.
         * For details see http://download.oracle.com/javase/tutorial/uiswing/lookandfeel/plaf.html 
         */
        try {
            for (javax.swing.UIManager.LookAndFeelInfo info : javax.swing.UIManager.getInstalledLookAndFeels()) {
                if ("Nimbus".equals(info.getName())) {
                    javax.swing.UIManager.setLookAndFeel(info.getClassName());
                    break;
                }
            }
        } catch (ClassNotFoundException ex) {
            java.util.logging.Logger.getLogger(ElAjedrez.class.getName()).log(java.util.logging.Level.SEVERE, null, ex);
        } catch (InstantiationException ex) {
            java.util.logging.Logger.getLogger(ElAjedrez.class.getName()).log(java.util.logging.Level.SEVERE, null, ex);
        } catch (IllegalAccessException ex) {
            java.util.logging.Logger.getLogger(ElAjedrez.class.getName()).log(java.util.logging.Level.SEVERE, null, ex);
        } catch (javax.swing.UnsupportedLookAndFeelException ex) {
            java.util.logging.Logger.getLogger(ElAjedrez.class.getName()).log(java.util.logging.Level.SEVERE, null, ex);
        }
        //</editor-fold>

        /* Create and display the form */
        java.awt.EventQueue.invokeLater(new Runnable() {
            public void run() {
                new ElAjedrez().setVisible(true);
            }
        });
    }

    // Variables declaration - do not modify                     
    private javax.swing.JButton jButton1;
    private javax.swing.JButton jButton2;
    private javax.swing.JPanel jPanel1;
    private com.stackoverflow.es.question371169.Tablero tablero1;
    // End of variables declaration                   
}

I did it with the help of netbeans, and it looks like this in matisse, which is the graphic editor it has:

Once the form is running, it looks like this:

With this solution you avoid having empty rows that are rendered , which is a problem that gridlayout has in certain particular conditions.