The monarch sent the Prime Minister a private message of good wishes and will also speak on the phone on Wednesday during their weekly audience.
The call is still scheduled to go ahead despite being due to take place just hours after the arrival of Mr Johnson’s baby boy.
Downing Street said Mr Johnson, who is currently grappling with the coronavirus crisis, would take a short period of paternity leave later this year.
Mr Johnson’s audiences with the 94-year-old Queen, who is staying at Windsor Castle, have been taking place weekly by telephone during lockdown.
They were put on hold while Mr Johnson was battling Covid-19 in intensive care earlier this month.
Foreign Secretary Dominic Raab – who deputised for the PM – was not asked to step in to brief the Queen on his behalf.
The Queen sent a message to Mr Johnson’s fiancee Ms Symonds and his family while he was being treated in hospital, wishing him a “full and speedy recovery”.
Mr Johnson is the 14th prime minister of the Queen’s reign.
Additional reporting by Press Association.