Zotero can be used with Google Docs in the same way as with plain-text documents or emails. See the screencast on Zotero and Google tools.
To use a different style, change your Quick Copy settings in the Export pane of the Zotero preferences. Here's how:
Select the citation style of your preference
In your Google doc leave the cursor at the end of your sentence(where you need your in-text citation) and hold the shift key before dragging the citation in the document then proceed to drop it in the Google doc.
To insert the bibliography, select the citations from your Zotero library and drag and drop in your Google doc, do not hold the shift key.
When dropped in the document, you should have a formatted bibliography in your Google doc.