I think the guide I made below will help to find pictures of Royals wearing their birth stones jewelleries.
Birth stones website.
Royal and their birthstones.
Queen Elizabeth II 26 April 1926Diamond, Turquoise and Emerald.
Princess Royal, Anne 15 August 1950Carnelian, Ruby, Tiger Eye and Peridot
Princess Alexandra was born on Christmas Day 1936 Ruby, Garnet and Amethyst
Queen Sonya Born 4 July 1937 Alexandrite, Ruby and Moonstone
Princess Märtha Louise born 22 September 1971 Sapphire, Rainbow Agate and Peridot
Crown Princess Mette-Marit was born on 19 August 1973, Carnelian, Ruby, Tiger Eye and Peridot
Queen Paola was born Princess Ruffo di Calabria, on 11 September 1937 Sapphire, Rainbow Agate and Peridot
Doña Fabiola, was born in Madrid on 11 June 1928 Amber, Emerald Moonstone and Lapis Lazuli
Princess Mathilde was born on 20 January 1973 in Uccle Garnet, Amethyst, and Ruby.
Princess Astrid was born in Brussels on 5 June 1962 Amber, Emerald Moonstone and Lapis Lazuli
Princess Claire was born in Bath, Somerset (England) on 18 January 1974 Garnet, Amethyst, and Ruby.
Princess Máxima 17 May 1971 Emerald, Diamond and Turquoise
Petra Laurentien Brinkhorst was born in Leiden on 25 May 1966 Emerald, Lapis Lazuli, Amber and Moonstone
Margriet Francisca was born on 19 January 1943. Garnet, Amethyst, and Ruby.
Marie-Hélène Angela (Marilène) was born on 4 February 1970 in Dieren. Blue Sapphire, Aquamarine, and Amethyst.
Princess Annette was born in The Hague on 18 April 1972Diamond, Aquamarine and Bloodstone
Su Majestad la Reina Doña Sofía nació en Atenas el 2 de noviembre de 1938 Alexandrite, Opal, Topaz and Carnelian.
Princesa de Asturias, nació el 15 de septiembre de 1972 en Oviedo, Sapphire, Rainbow Agate and Peridot
Infanta Doña Elena, Nació en Madrid el 20 de diciembre de 1963, Ruby, Topaz, Turquoise.
Infanta Doña Cristina, Nació en Madrid el 13 de junio de 1965 Amber, Emerald Moonstone and Lapis Lazuli
Serbia and Yugoslavia RF
HRH Crown Princess Katherine was born in Athens, Greece13 November 1943 Alexandrite, Opal, Topaz and Carnelian .
Queen Margrethe II was born on 16 April 1940 Diamond, Aquamarine and Bloodstone
Princess Mary, was born on 5 February 1972Blue Sapphire, Aquamarine, and Amethyst
Princess Benedikte was born on 29 April 1944 Diamond, Turquoise and Emerald.
Princess Elisabeth was born on 8 May 1935. Emerald, Diamond and Turquoise
Princess Alexandra Christina was born on 30 June 1964 Ruby, Alexandrite and Moonstone.
Feel free to add more member RF from the guide above or add more information from other royal house