it's very interesting, I had never seen this brooch before.
I can not be 100% sure, but my feeling is that the emerald brooch worn by the Queen Mother is not one of Queen Mary's wedding gifts. Reason for this impression is that the style of the brooch looks too modern to be a gift from the late 19th century. But this is only my personal visual impression.
The picture you have posted is dated by Getty Pictures 3rd January, 1996 and depicts Queen Mother Elizabeth at the Grand Military Gold Cup at Sandown Park Racecourse in Esher, Surrey. Note that the picture has been flipped left-right by Getty pictures (as you can see on the 2nd picture below, the brooch is worn on the left lapel of the Queen-Mother's coat.
Caption: The Queen Mother attends the Grand Military Gold Cup. Held annually at Sandown Park Racecourse in Esher, Surrey it is a meeting point for the Military, in particularly for Cavalry Officers, with its origins in the days when mounted Cavalry Officers still rode to war. (Photo by Richard Manning/Getty Images)
I wonder if the emerald shamrock pin presented as a wedding gift to Queen Mary by Sir Frederick & Lady Milner could be this brooch worn by Princess Marina, Duchess of Kent, and later by Princess Michael of Kent: