A number of surviving Carnteel parish Methodist records are available on microfilm at PRONI. See their guide:
www.proni.gov.uk/guide_to_church_records.pdfIf you have Thomas John Davidson's 1860 birth/baptism record (as the exact date and place suggests), then you should know his parents' names, per that record, correct? Or maybe you only have his father's name on that record, and so are doubtful of the mother? If you don't have this actual baptism record and only found it cited somewhere else as a date & place, PRONI's microfilmed Methodist baptisms start in 1877 for Carnteel parish so won't have an 1860 baptism.
Non-established church denominations like Methodists often recorded vital events in the established Church of Ireland, too, so you should check that church in Carnteel parish, and it does include 1860 baptisms.
Methodist archivist Dr. Robin Roddie might be able to help you, and although he's not a commercial researcher, I'd send him a payment if I asked him to help with questions on Carnteel parish records:
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Depending on how interested you are in finding out more on this family and where you live, you can do PRONI research yourself or you can have a professional researcher check those records for an hourly fee.
Good luck with your search,
Annie
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