What actually happened was that The Sun ran a provocative piece on 9 March 2016, claiming that the Queen backed Brexit. The assertion was that she had “rip at the then Deputy PM [Nick Clegg] during a lunch at Windsor Castle”. Clegg rapidly tweeted “As I told the journalist this is nonsense. I’ve no recollection of this happening & its not the sort of thing I would forget” and the BBC ran a story about Buckingham Palace being clear that she was neutral over Brexit.
As the lunch in question was in 2011 it is stretching things to construe whatever was actually said as a comment on the forthcoming referendum. At this distance it might also be quite hard to be sure of what was said and what people wish she had said.
I don’t pretend to know the Queen’s private views on this, but the whole episode raises someone in the role of the Queen might respond to the EU, in terms of the role itself, history and the constitution.