Can The Queen Sack Boris Johnson?

A question i have been asked three times now is can the Queen remove Boris Johnson, the man we are referring to, for now at least, as the Prime Minister?
I always answer that i doubt it due to antiquated rules established around the time of the Magna Carta and the whole Oliver Cromwell thing but i thought i had better checked so i looked up the House of Commons Library which say's no she can't sack him but there was a but...
While the Prime Minister may be head of the government appointed by The Queen, there are certain residual powers retained by the monarch from bygone days where if the UK Prime Minister loses a vote of confidence in the house, and then 14 days later loses another one, it should sound the starting whistle on a general election and the matter is put to the electorate to resolve.
If after the two votes of no confidence, if the Prime Minister refuses to resign, then the Queen can't sack him but can invite another political leader from the same party or the shadow Government to try to form a government and then if they win a vote of confidence from the members in the Houses of Commons, the Queen swears in the new Prime Minister replacing the old one.
This leaves the ultimate power of appointing and dismissing Prime Ministers in the hands of the elected representatives in Parliament and the Queen therefore can't do anything until a second vote of No Confidence has been tabled and succeeded and only then if Boris refuses to pack up his bags and leave Downing Street.
It is unlikely the Labour Party will pushing for a vote of no-confidence, it is in their favour to leave a hugely unpopular leader on the opposition benches hemorrhaging votes so it will probably come from the Conservative Party itself and the 1922 Committee who can call a vote of no-confidence if 15% of Conservative MPs (54) send a letter requesting one.
This will stop a General Election they would probably lose and keep a Conservative Government in place until at least the next election in 2024.