Unlike Word there isn't Cite While You Write functionality between EndNote and Google Docs. However, you can insert unformatted citations from an EndNote library into a Google Doc, save it as RTF file and then format the document using EndNote and create a document that will include citation and references in the selected publication style.
Highlight the appropriate reference(s) in your EndNote library and drag and drop the reference(s) into the Google doc where the citation is needed This will create an unformatted citations within your Google doc.
Once all the citations have been entered into your Google doc download the document as an RTF file.
Go to Endnote library, select Tools > Format Paper > Format Paper and select your unformatted .rtf file
In the Format Paper box, select the Output style (e.g. JAMA, APA, etc.) and then select Format.
Save the newly formatted file and choose where to save it. Do not load this version back into Google Docs or the formatting will disappear.
You can now open your document in Word with the properly formatted citation and references.