Before upgrading, do I need to upgrade the Views module to 6.3.x? I'm going to update a site from Drupal 6 to Drupal 7. It uses the Views module version 6.x-2.

I would like to know if I need to upgrade the Views module to version 6.x-3 before moving to Drupal 7.
It's not necessary. I have migrated one large site from Drupal6+Views2 to Drupal7+Views3 in just one step.

You have to take in account that views3 (views_ui submodule) depends on ctools. If you follow this path you have to disable views_ui before the upgrade and activate views_ui (and ctools) once the upgrade has finished.