Here’s a link to the 1882 death. That Ezekiel was a shoemaker so given the comparative rarity of both name and occupation, I’d suspect it’s the right man.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1882/06416/4842680.pdf
He was a widower, lived in Ballynaloob and the informant was Martha Greer, of the same townland.
I searched for Ann Grier’s death 1864 to 1882 but did not find it. So it’s possible she died before 1864 which is when death registration began.
This might be Martha in 1901. If so her age would make her possibly Ezekiel's daughter:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Killagan/Drumadarragh/944425/
She is lodging with John Greer & his wife Jane Stewart. Their son Stewart was born in Ballynaloob in 1895:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1896/02186/1827119.pdf
John Greer’s father was William Greer, so no direct link to Ezekiel but given they were in the same townland, I suspect there’s a connection:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1883/10938/5997239.pdf
This looks to be Ezekiel Greer’s daughter Sarah who married in 1867:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1883/10938/5997239.pdf
Sarah was living in Ballyoglagh, parish of Billy, at the time. This looks to be her in Ballymena in 1901. She’s probably been a little economical with her age:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Ballymena_and_Town_and_Urban_District_of_Ballymena/Clonavon_Place_S_/925858/
Same family in 1911:
http://www.census.nationalarchives.ie/pages/1911/Antrim/Ballymena__No__2__Urban__In_34_files_/Clonavon_Road/117409/
They had 4 children all of whom were alive in 1911. Birth registration only started in Ireland in 1864 so you won't find birth certificates for Ezekiel, his wife or their children. You might find baptisms but to do that you would need to know which church they were baptised in. The records are mostly not on-line. The most complete collection is in PRONI (the public record office) in Belfast but there's a lot of churches in the area around Ballynaloob.
