If you own a character account, and do not want to share the data to public, you can change private settings on Lodestone, and a private flag will be applied on next update, preventing any further updates, hiding all details of the profile and applying noindex tag to the page. Note, that the update will follow the sync flows described here. In short this means, that there will most likely be a delay in the update. Character pages do show when they were last updated, though, and once the update is applied, you will see a significant difference.
Note, that you may need to force-refresh the page (clear browser cache) to see the change. This also does not imply instant removal of the page from Google search results: the noindex tag is meant to help with that, but that's all it does: help. In addition link to profile may still be present on groups' pages, same as with Lodestone, and the data will still be stored in database.
If you want to do the reverse, that is to start showing the data on tracker, then you should make your profile public for everyone, and let the page be updated as per the same flows. Note, that you do not need to make everything public, just the main profile, that is Character Information -> Profile setting.
