Wikiquote:Deletion process
An article's VfD discussion page has a URL such as Wikiquote:Votes for deletion/PAGENAME.
An admin closing a discussion at VfD uses two of Wikiquote's archival templates on the article's VfD discussion page: Template:Vfd top and Template:Vfd bottom.
Edit the discussion page, putting {{vfd top|result}} ~~~~ at the top of it and {{vfd bottom}} at the bottom of it, replacing the word "result" with a brief statement of the result.
If the consensus was to delete, the process is much what it would be for a speedy or PROD deletion. Before deleting, check for other articles that link to the article, including checking its Wikipedia article for a Wikquote link. After making sure you aren't breaking anything, delete the article and its talk page, if any.