Google Docs support is part of the Zotero Connector for Chrome and Firefox. 

To start make sure that you have installed the Zotero connector for Chrome or Firefox. 

Start typing your paper. When you need to add a citation, click on the "Zotero" tab and then "Add/edit citation" option.

Select the appropriate style format for your paper (Zotero will only ask you once, for each document. To change this later, go to the Zotero tab and then Document preferences).

A pop-up Zotero search window will appear. You can search for your selected source by typing in a keyword (author name or word from title etc.)

Note your citation has been inserted.

Zotero will also automatically generate a list of your references if you click on the "Add/edit Bibliography" option. You only need to click on this icon once. From now on, each time you add a citation, Zotero will automatically add references to your bibliography in the proper order and in the proper style format.

The Quick Copy or "drag and drop" method works for both MS Word and Google Docs. 

Add In-Text Citations: Hold down the shift key, select a reference from the middle pane of Zotero, and (still holding down the shift key), drag it into Word or Google Docs.

Add Bibliography: Using the CTRL key (Windows) or CMD key (Mac), select all the references you wish to include and drag them into your paper.