. . . . . . . . Queen of England lives in . . . . . . . . Buckingham Palace in London.

Option C is correct because 'The' is used in the first blank to provide the sense to the sentence. 'The' refers to a particular person & shows the uniqueness of the subject of which the readers are aware of. The Queen of England is a specific person.
The second blank will take 'no article' as Buckingham Palace is a proper noun and most proper noun does not take any article.