Your problem may well be the one that all people tracing their Irish ancestry eventually face, i.e. once you get beyond registers of baptisms/births and marriages (which are the building blocks of Irish family history) you have few means (unless the precise family history has been documented and passed down through the generations) to confirm family connections.
It would appear that you should concentrate your initial research in Ballymoney parish, County Antrim as the townland of Conagher (spelt as Conogher in Townland Index) is located in this parish.
You can identify all church registers in Ballymoney parish (and indeed in all parishes in Northern Ireland), together with commencement dates, by examining the church guide produced by Public Record Office of Northern Ireland, https://www.nidirect.gov.uk/publications/guide-church-records
It would appear that only one church in Ballymoney parish has surviving registers prior to 1800, namely 1st Ballymoney Presbyterian church which has baptism registers for the period 1751-1771.
With best wishes