I found a work around, keeping in mind that the pager works for calendar view by changing date in the URL.

I created a custom form with fields as day, month, year whose field type is select (dropdown).

This is the code for the form and the submission handler.

    function mymodule_get_event_view_date_filter($form, &$form_state) {
      $form = array();
      $year = date('Y');
      $year_options = array(
        $year - 4 => t($year - 4),
        $year - 3 => t($year - 3),
        $year - 2 => t($year - 2),
        $year - 1 => t($year - 1),
        $year => t($year),
        $year + 1 => t($year + 1),
        $year + 2 => t($year + 2),
        $year + 3 => t($year + 3),
        $year + 4 => t($year + 4),
      );
      $month = date('m');
      $month_options = array(
        '1' => 'Jan',
        '2' => 'Feb',
        '3' => 'March',
        '4' => 'April',
        '5' => 'May',
        '6' => 'June',
        '7' => 'July',
        '8' => 'Aug',
        '9' => 'Sep',
        '10' => 'Oct',
        '11' => 'Nov',
        '12' => 'Dec',
      );
      $form['event-date-container'] = array(
        '#type' => 'container',
        '#prefix' => '<div id="event-date-container">',
        '#suffix' => '</div>',
      );
      $form['event-date-container']['year'] = array(
        '#type' => 'select',
        '#options' => $year_options,
        '#prefix' => '<div id="event-year">',
        '#default_value' => isset($_SESSION['event']['year']) ? $_SESSION['event']['year'] : $year,
        '#suffix' => '</div>',
      );
      $form['event-date-container']['month'] = array(
        '#type' => 'select',
        '#options' => $month_options,
        '#prefix' => '<div id="event-month">',
        '#default_value' => isset($_SESSION['event']['month']) ? $_SESSION['event']['month'] : $month,
        '#suffix' => '</div>',
      );
      $form['event-date-container']['submit_form'] = array(
        '#type' => 'submit',
        '#value' => t('Apply'),
        '#submit' => array('_mymodule_event_view_date_submit'),
      );
      unset($_SESSION['event']);
      return $form;
    }
    
    function _mymodule_event_view_date_submit($form, $form_state) {
      global $base_url;
      $_SESSION['event']['year'] = $form_state['values']['year'];
      $_SESSION['event']['month'] = $form_state['values']['month'];
      drupal_goto('calendar-node-field-event-date/month/' . $form_state['values']['year'] . '-' . $form_state['values']['month']);
    }

Now call this form in your calendar view in view header print.

    drupal_render(drupal_get_form(mymodule_get_event_view_date_filter));