The below code worked for me. Thanks @Abra.

import java.util.Date;
import javax.swing.JSpinner;
import javax.swing.SpinnerDateModel;

public class SpinnerDatesExample extends javax.swing.JFrame {

    private javax.swing.JButton jButton_Update;
    private javax.swing.JSpinner jSpinner_EndDateTime;
    private javax.swing.JSpinner jSpinner_StartDateTime;

    private SpinnerDateModel StartDateModel, EndDateModel;

    private Date StartDateTime, EndDateTime;
    //private boolean flag = false;

    public SpinnerDatesExample() {
        initComponents();
        {
            jSpinner_StartDateTime.setEditor(new JSpinner.DateEditor(jSpinner_StartDateTime, "yyyy-MM-dd hh:mm:ss.SSS"));
            jSpinner_EndDateTime.setEditor(new JSpinner.DateEditor(jSpinner_EndDateTime, "yyyy-MM-dd hh:mm:ss.SSS"));
        }
        {
            StartDateModel = (SpinnerDateModel) jSpinner_StartDateTime.getModel();
            EndDateModel = (SpinnerDateModel) jSpinner_EndDateTime.getModel();
        }
        updateDates();
    }

    private void updateDates() {
        do {
            StartDateTime = new Date((2020 + (int) Math.round(Math.random() * (2025 - 2020))),
                    (1 + (int) Math.round(Math.random() * (12 - 1))),
                    (1 + (int) Math.round(Math.random() * (27 - 1))),
                    (0 + (int) Math.round(Math.random() * (23))),
                    (0 + (int) Math.round(Math.random() * (59))),
                    (0 + (int) Math.round(Math.random() * (59))));

            EndDateTime = new Date((2020 + (int) Math.round(Math.random() * (2025 - 2020))),
                    (1 + (int) Math.round(Math.random() * (12 - 1))),
                    (1 + (int) Math.round(Math.random() * (27 - 1))),
                    (0 + (int) Math.round(Math.random() * (23))),
                    (0 + (int) Math.round(Math.random() * (59))),
                    (0 + (int) Math.round(Math.random() * (59))));
        } while (StartDateTime.after(EndDateTime));

        {
            System.out.println("StartDateTime -> " + StartDateTime);
            System.out.println("EndDateTime -> " + EndDateTime);
        }

        //  Alternatively use different methods I used that did not fulfill my requirements.
        //  This will demonstrate my issue.
        /*if (flag) {
            jSpinner_StartDateTime.setModel(new javax.swing.SpinnerDateModel(StartDateTime, StartDateTime, EndDateTime, java.util.Calendar.SECOND));
            jSpinner_EndDateTime.setModel(new javax.swing.SpinnerDateModel(EndDateTime, StartDateTime, EndDateTime, java.util.Calendar.SECOND));
            flag = false;
        } else {
            jSpinner_StartDateTime.getModel().setValue(StartDateTime);
            jSpinner_EndDateTime.getModel().setValue(EndDateTime);
            flag = true;
        }*/
        {
            StartDateModel.setStart(StartDateTime);
            StartDateModel.setEnd(EndDateTime);
            StartDateModel.setValue(StartDateTime);
        }
        {
            EndDateModel.setStart(StartDateTime);
            EndDateModel.setEnd(EndDateTime);
            EndDateModel.setValue(EndDateTime);
        }
    }

    private void initComponents() {

        jSpinner_StartDateTime = new javax.swing.JSpinner();
        jSpinner_EndDateTime = new javax.swing.JSpinner();
        jButton_Update = new javax.swing.JButton();

        setDefaultCloseOperation(javax.swing.WindowConstants.EXIT_ON_CLOSE);

        jSpinner_StartDateTime.setModel(new javax.swing.SpinnerDateModel(new java.util.Date(), null, null, java.util.Calendar.HOUR_OF_DAY));
        jSpinner_EndDateTime.setModel(new javax.swing.SpinnerDateModel(new java.util.Date(), null, null, java.util.Calendar.HOUR_OF_DAY));

        jButton_Update.setText("Update Dates");
        jButton_Update.addActionListener(new java.awt.event.ActionListener() {
            public void actionPerformed(java.awt.event.ActionEvent evt) {
                jButton_UpdateActionPerformed(evt);
            }
        });

        javax.swing.GroupLayout layout = new javax.swing.GroupLayout(getContentPane());
        getContentPane().setLayout(layout);
        layout.setHorizontalGroup(
                layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
                        .addGroup(layout.createSequentialGroup()
                                .addContainerGap()
                                .addGroup(layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
                                        .addComponent(jSpinner_StartDateTime, javax.swing.GroupLayout.DEFAULT_SIZE, 533, Short.MAX_VALUE)
                                        .addComponent(jSpinner_EndDateTime)
                                        .addGroup(javax.swing.GroupLayout.Alignment.TRAILING, layout.createSequentialGroup()
                                                .addGap(0, 0, Short.MAX_VALUE)
                                                .addComponent(jButton_Update)))
                                .addContainerGap())
        );
        layout.setVerticalGroup(
                layout.createParallelGroup(javax.swing.GroupLayout.Alignment.LEADING)
                        .addGroup(layout.createSequentialGroup()
                                .addContainerGap()
                                .addComponent(jSpinner_StartDateTime, javax.swing.GroupLayout.PREFERRED_SIZE, javax.swing.GroupLayout.DEFAULT_SIZE, javax.swing.GroupLayout.PREFERRED_SIZE)
                                .addPreferredGap(javax.swing.LayoutStyle.ComponentPlacement.RELATED)
                                .addComponent(jSpinner_EndDateTime, javax.swing.GroupLayout.PREFERRED_SIZE, javax.swing.GroupLayout.DEFAULT_SIZE, javax.swing.GroupLayout.PREFERRED_SIZE)
                                .addPreferredGap(javax.swing.LayoutStyle.ComponentPlacement.RELATED)
                                .addComponent(jButton_Update)
                                .addContainerGap(javax.swing.GroupLayout.DEFAULT_SIZE, Short.MAX_VALUE))
        );
        pack();
    }

    private void jButton_UpdateActionPerformed(java.awt.event.ActionEvent evt) {
        updateDates();
    }

    public static void main(String args[]) {
        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 | InstantiationException | IllegalAccessException | javax.swing.UnsupportedLookAndFeelException ex) {
            java.util.logging.Logger.getLogger(SpinnerDatesExample.class.getName()).log(java.util.logging.Level.SEVERE, null, ex);
        }

        java.awt.EventQueue.invokeLater(new Runnable() {
            public void run() {
                new SpinnerIssueExample().setVisible(true);
            }
        });
    }
}