There are no birth certificates on-line for the 1920s (there is a 100 year restriction) but there are marriage and death certificates available (marriages are closed for 75 years, deaths for 50). The “closed” records are open to the public, just not on-line unless you can get to PRONI or GRONI in person. The terminals there allow you to search all events right up to those registered yesterday.
There is a death registered in Ballymoney for Annie Quinn on 31.1.1927 aged 0. So that fits the information you have given pretty accurately. You can view the original certificate on-line on the GRONI website, using the “search registrations” option:
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
If you want a copy of the matching birth certificate, you can e-mail GRONI with the details and approximate year of birth (obviously 1926/7) and they will send you a copy, for a fee. Contact details for e-mailing GRONI:
As to where the family were living, it’s looks as though it wasn’t always Killyramer and they moved around a little though maybe not very far. Here’s Thomas’s birth certificate for 1917 when they were living at Dungorbery (Kilraghts). The father John was working as a coal miner in Motherwell (Scotland) at that date.
Here’s John & Nancy’s marriage in 1916 when he was living in Ballyboyland. Nancy was in Dungorbery then. (Dungorbery and Killyramer are beside each other. Moving from one to the other probably involved a journey of 200 yards).
I looked in the Valuation Revision records for Killyramer on the PRONI website. They show owners and tenants of properties. The records for 1913 - 1930 show one Quinn household. That was Betty Quinn who had a cottage (plot 4c) on land owned by John Megaw. Here’s that household in the 1901 & 1911 censuses:
The above family was Presbyterian.
There were only 18 houses in Killyramer in 1911, and just the one Quinn so I presume there’s a connection. The location of Betty Quinn’s cottage today is close to the modern Gracehill Rd. There was once a lane that led off that road up to the farm and cottages. Looking at the site today on Google Earth all the buildings have gone and it is just one big field now, though if you look carefully you can see the outline of where the lane and properties were.
The gravestone transcriptions for Kilraghts Presbyterian and Kilraghts Reformed Presbyterian church are both on-line on the Causeway Coast & Glens Family History site. (Membership required). There are no Quinn gravestones in either churchyard. However she could be buried there without a gravestone in which case it may be tricky locating the grave. You would need to contact the churches to see if they have burial records for the 1920s (Many don’t. Presbyterians generally didn’t keep burial records). There are also graveyards nearby at Kilraughts Old Graveyard and Bushvale Presbyterian church. No Quinn headstones in Bushvale. The Old Kilraghts gravestones don’t appear to be on-line. These graveyards are all fairly close to Killyramer.