A Quaker meeting was established in Ballinacree / Ballynacree near Ballymoney around 1673 [Guide to Irish Quaker records page 413]. The Moore family were said to have been closely involved with this meeting and I take James Moore [PRONI will 1727] to be the progenitor of the family. The main meeting place was in Antrim so you will find records relating to Ballynacree in the Antrim minutes in PRONI ref. MIC16/43:
ANTRIM OR GRANGE MONTHLY MEETING [GRANGE, BALLYNACREE, TOBERHEAD, COLERAINE AND ANTRIM]
7 Feb. 1740-4 Sept. 1763 Men's minutes (rough). [Antrim or Grange.]
16 Oct. 1763-24 May 1778 Men's minutes. [Antrim or Grange.]
10 Sept. 1780-13 Nov. 1785 Men's minutes (rough). [Antrim or Grange.]
25 Dec. 1785-1 Jan. 1790 Men's minutes (rough). [Antrim or Grange.]
12 Feb. 1790-12 Aug. 1791 Men's minutes (rough). [Antrim or Grange.]
23 Sep. 1791-8 Feb. 1793 Men's minutes (rough). [Antrim or Grange.]
12 Apr. 1793-9 May 1794 Men's minutes (rough). [Antrim or Grange.]
6 June 1794-7 Sep. 1798 Men's minutes (rough). [Antrim or Grange.]
12 Oct. 1798-9 May 1801 Men's minutes (rough).
11 June-12 Nov. 1801 Men's minutes (rough).
9 May 1794-8 Nov. 1800 Women's minutes [Antrim or Grange.]
1741-1800 Volume containing testimonies of disownment, letters of condemnation, certificates of removal, 1758-1800, marriages, 1768-77, births, 1751-1800, and deaths, 1741-97, and details about the discontinuance of Toberhead meeting, 1796.
IMMIGRATION OF IRISH QUAKERS INTO PENNSYLVANIAImmigration of the Irish Quakers into Pennsylvania, 1682-1750 : with their early history in Ireland (1902) by Albert Myers
This book mentions that in the period 1682-1750 local Quaker meetings in Pennsylvania received 35 persons from the Ballynacree meeting with certificates of approval. Many of the certificates are quoted in the book which can be downloaded for free [searchable PDF] in the link below:
http://archive.org/details/immigrationofiri00myerThe O'Mooney family is mentioned in this volume [pages 34; 160; 354; 366].
Myer also mentions in his book that he visited Ballymoney in the year 1900 looking for evidence of where the Ballynacree meeting had been held but the knowledge had passed out of local folklore. Someone was able to tell him the of the Quaker burial ground for Ballynacree which he found out was at 'Lamb's Fold at Enogh near O'Hara's Brook about 2 miles west of Ballymoney'. He does not mention if headstones had been erected.