At the moment Mary Head is said to have been born in Edmondsham, Dorset with parents William Head and Mary Cutler. The source is a baptism of a Mary Head in Edmondsham in 1773.
These are the research notes I've added (references and image of gravestone on profile) The profile is orphaned so I have no-one to discuss this with.
According to Mary's gravestone, she was born in 1772, her father was stated to be James Head who came from Fawley, Berkshire and who died in 1772 . Her mother was said to be Mary. In the 1851 census where her age was given as 77, her birthplace was entered as Reading, Berkshire [11]
According to references describing Hardy's Wessex, Marygreen in Jude the Obscure refers to Fawley in Berkshire where his grandmother was born. He visited Fawley, presumably in search of his roots. [12] [13]
In addition I've found a facebook post which mentions his grandmother's baptism in the Fawley register, daughter of James and Mary (Hobson) [don't know if this link will work https://www.facebook.com/royalberkshirehistory/posts/one-of-the-people-ive-been-researching-recently-is-mary-head-the-paternal-grandm/2481994528531051/
I'm not able to find this baptism, suspect the register might be on Findmypast which I don't have access to at the moment. There is a marriage of a James Head and Mary Hobson in 1771 in Wantage, Berkshire https://ukga.org/search.php?action=ViewRec&DB=36&recid=7944