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RC records for Kilbarron RC parish do not start until the 1850s so no records exist for the Quigleys in your message.
I checked the Tithe listings for Kilbarron parish and there were no Quigleys listed.
The Mary Quigley in the Griffiths is likely the mother of James. I did look for a death record index record for a Mary Quigley in the Ballyshannon registration district but did not find a record.
You may want to add James' story to our XO Chronicles site https://www.irelandxo.com/ireland-xo/history-and-genealogy/ancestor-data...
You may also want to consider autosomal DNA testing.