Can you provide a screenshot or two of what you're talking about? It sounds like perhaps there's a web content widget on a Smartsheet dashboard that is displaying a Google Docs document, and inside that document there's a checkbox?

There's no data connection between Smartsheet and Google Docs. So if my summary above is correct, then yes you would not expect edits inside Google Docs to have any effect on your Smartsheet data. The only data connection is one way outbound from Smartsheet to Google Docs as part of the integration that creates a new google doc using Smartsheet data.