The Zotero Google Docs function is part of the Zotero connector for Chrome, Firefox, Edge, and Safari. It also requires the Zotero standalone program.
When you first interact with Zotero for the first time you will be prompted to authenticate the plugin with your Google account. Be sure to select the Google account you used to create the document or that has been given editing access by the document's creator. After authentication, you can insert citations from your Zotero libraries.
For more information consult Using Zotero with Google Docs
Downloading and installing Zotero on your laptop or desktop will automatically include Zotero's MS Word Plugin. After installing and running Zotero open MS Word you should see the Zotero Tab in the MS Word ribbon.
For additional information on using Zotero in MS Word see Using the Zotero Word Plugin