David,
A small portion of the 1851 Irish census has survived and it includes part of Killead. It lists 1 McConkey household in Ballyquillan, plus another where a McConkey was working as a servant. No obvious link to your family but they were living near Martha and so might be related. (I noticed that a Hugh McConkey was a witness to the 1867 marriage in Killead, so wonder if this might be the same Hugh. A cousin perhaps?)
http://www.census.nationalarchives.ie/pages/1851/Antrim/Lower_Masserene/Killead/Ballyquillan/4/
http://www.census.nationalarchives.ie/pages/1851/Antrim/Lower_Masserene/Killead/Ballyquillan/5/
Family was Church of Ireland judging by the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Seacash/Ballyquillan/917528/
Hugh McConkey was married to Mary Ramsay:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1869/03387/2241555.pdf
She died in 1878:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1878/020504/7195256.pdf
Hugh married in 1867. His father was James a farmer.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1867/11508/8232124.pdf
Here’s James probate abstract:
The Will of James M'Conkey late of Ballyquillan in the County of Antrim Farmer deceased who died 29 November 1865 at same place was proved at Belfast by the oaths of William John M'Conkey of Ballyquillan and Alexander Nixon of Ballynadrenta both in (Crumlin) the said County Farmers two of the Executors.
The will itself is on-line on the PRONI site and mentions James’ wife Mary, son William John who was to inherit 1 farm in Ballyquillan, son Hugh who was to inherit a second (Jones Land) and son David who was to inherit a 3rd situated in Crosshill. there was also provision for 2 unmarried daughters Margaret & Anne Jane.
Daniel’s marriage here:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
The portion of the 1851 census for Tully has not survived, sadly so we don’t know if Martha McConkey was in it.
Marriage here for a David McConkey of Killead in 1852. Father was Robert, a labourer. Is that a brother to your Thomas?
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1852/09420/5412582.pdf
As you probably know, there are 2 Church of Ireland churches in Killead (Gartree & Killead). Both churches early records were lost in the 1922 fire. Gartree has nothing before 1899 now and Killead nothing before 1838. There were 2 Presbyterian churches in the parish (tradition being to marry in the bride’s church. Note that Thomas’s second marriage was in Killead Church of Ireland, not St Anne’s in Belfast). Killead Presbyterian has no records earlier than 1826 and Loanends nothing before 1834.
Robert & Martha evidently married before 1816. Sadly there are no records that might show their marriage or their own baptisms, for the Killead area. Getting back any earlier will be challenging.
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).