When you open Google Docs in a browser that has a Zotero connector installed, you will see a Zotero option added to your menu.
Interacting with the Zotero functionality for the first time in a document will prompt you to authenticate the plugin with your Google account. Once authenticated you can begin inserting citations from your Zotero libraries just as you can with Word and LibreOffice.
To add a citation to your document, you can choose "Add/edit citation" from the menu or click on the Z button in the Google Docs toolbar. You will be prompted to choose your citation style. Choose the citation using the citation dialog. Add citations and create bibliographies as described under the Bibliographies tab of this guide.
There are only a few differences in how Google Docs works with Zotero from how it works with Word and LibreOffice:
One advantage of using Zotero with Google Docs is that multiple people can collaborate on a single document -- inserting and editing citations -- without being in a Zotero group. However, if someone cites an item from their personal library, only they will be able to update the metadata for that item. For large projects, Zotero recommends creating a group library that allows all collaborators to change cited item metadata.
It is not necessary for all collaborators to have Zotero installed, but if they do not have the Zotero connector installed in the browser they use to view the Google Doc, they may break the citation links from the document to the Zotero library.