Zapier.com is an automation website. Users can build a chain of events called zaps based on things that happen.
In your case, you would have a zap that states "on any new activity in Trello board XZY, append Google Drive document with the date time and description of that activity."
It's actually quite simple to build. The website isn't built for tech experts. If you follow the instructions on the site, you should be able to quickly create the zap you need.