From The Times (of London) April 22, 1926
and from the Court Circular
By pure coincidence, on the day of Elizabeth's birth her future mother-in-law, Princess Andrew (Alice) of Greece, along with her younger sister, the Crown Princess of Sweden, and their mother, the Dowager Marchioness of Milford Haven (nee Princess Victoria of Hesse and by Rhine) visited King George V and Queen Mary at Buckingham Palace and stayed for lunch. One wonder if Princess Andrew's nearly 5-year-old son, Prince Philip, had remained at Kensington Palace with his nanny. The Dowager Marchioness lived in an apartment at Kensington Palace. She was the King's first cousin.
It is possible that Queen Mary may have asked Princess Alice about her young son. But on April 21, 1926, the guests may have offered a toast in honor of the new princess, it is unlikely that the King and Queen and their guests would have thought, let alone considered the possibility that the young Philip would become the yet-to-be-named infant princess's first and only love.