Tara Judy,
Irish marriage certificates only record the townlands (addresses) of the 2 parties, not where their fathers live. So though Jemima was living in Springmount, it doesn’t follow her father was. And he certainly isn’t listed there in Griffiths Valuation (1850). Springmount looks to be just one big 165 acre estate belonging to the March family, so I assume Jemima was working there as a governess or servant. That could account for her move from Tyrone to Queens.
Until comparatively recent times, Irish statutory marriage certificates only recorded the fathers names. Mothers weren’t noted.
Farmers didn’t move around much if they could avoid it, and so if Jemima was born in Tyrone, then I’d expect Arthur to be in Tyrone too. That said there’s no Arthur Green(e) in Tyrone in Griffiths Valuation for 1860. Nor is there one in the tithe applotment records for c 1830. I searched the death records 1864 to 1911 but did not find one in Tyrone either. It’s not a common name and there are not that many across the whole of Ireland. 1901 census had 16 Arthur Greens in the whole of Ireland. None in Tyrone (or Queens), save for 1, an unmarried servant born in England:
http://www.census.nationalarchives.ie/pages/1901/Queen_s_Co_/Emo/Emo_Park/1648161/
Jemima gave her age as 22 on the marriage certificate so her birth is well before the start of statutory birth registration in Ireland (1864). Prior to that you need to rely on church baptism records, not all of which are on-line. Tradition was to marry in the bride’s church which in this case was the local Methodist church, so she presumably was Methodist at least until she married. So you are probably looking for a Methodist baptism in Co. Tyrone. That said, some Methodists still used the Church of Ireland (COI) then, so it’s possible she might be baptised COI. Not many Methodists baptism records in Ireland are on-line. There are copies of most sets of records for Tyrone in PRONI (the public record office) in Belfast but a personal visit is required to view them. And there are probably quite a few Methodists chapels in Tyrone. Plus you might need to search the COI ones too. Quite a daunting task.
You could look at Griffiths Valuation and see where the surname Green(e) was found in Tyrone and focus on those parishes. There’s not that many, perhaps 8 or 9 parishes, so that could narrow the search area.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).