Hi Manick,
Can't say much about cricket in Dunfermline, and nothing about Queen Anne School, although now-deceased relatives of mine went there, I believe.
Dunfermline's Cricket Club was started in 1857, and they leased their own cricket field at Ladysmill at the south end of the town from 1879. In 1906 this field was bought by a local man, John McKane, who'd emigrated to the USA and become a millionaire. He presented the field to the Club, which was then named McKane Park. It is still (or was, fairly recently) used for cricket matches, though I've heard of a scheme to have it sold off for housing "development".
Dunfermline CC seem to have had their best seasons in the 1930s, winning various championships.
Maybe Queen Anne played at McKane Park? Pity your photo doesn't show more background!
Harry.