Hi C,
In 1851 there was only one William Stone living in Quarndon. He was 12 and living with the Gallimore family working there as a servant. There was a Sarah Stone living in Quarndon with 5 children. It is likely she was William's mother as she was the only Stone living in Quarndon that was of an appropriate age (she would have given birth to him when she was around 23). Also, William would have likely been her eldest child, and her being recently widowed would explain why he was living elsewhere and working as a servant (needing to make ends meet!).
I also found him in 1841 aged 3 living with his father George, mother Sarah and sister Jane (aged 1).
I hope this helps! If there is anything else I can help with just let me know.
All the best,
Ashlee