Is the Queen referred to incumbent Queen Elizabeth? Who was the King? What is the difference between these two kind of English?
The term "the Queen's English" is used when a queen is on the throne of England so you are right - at the moment it is Queen Elizabeth II. The term used whenever there is a king on the throne is "the King's English" but the English is the same it's only the name that changes.
Is the the accent of BBC news Queen's English?
Previously yes, but these days we can hear a variety of accents from many parts of Britain on the BBC. The term "the Queen's English" refers to standard southern English.