After hours of research, I've finally come up with a default core solution that I'd like to share with everyone on how to translate the Contact form module that comes with Drupal in Drupal 6.x.
Basic Instructions:
Make sure your Contact form module is enabled first, of course.
Go to drupal\modules\contact and open the file contact.pages.inc.
Go to line 41.
Replace the code...
$form['contact_information'] = array('#value' => filter_xss_admin(variable_get('contact_form_information', t('You can leave a message using the contact form below.'))));
...with:
$form['contact_information'] = array('#value' => t(filter_xss_admin(variable_get('contact_form_information', 'You can leave a message using the contact form below.'))));
Once replaced, save the page.
Next, install the module String Overrides.
Once installed, go to the String Overrides main page on your Drupal site under admin/settings/stringoverrides and click on the preferred language you would like to translate your contact form text to.