Uriah, a non-Quaker, married Mary Elizabeth Cook, a Quaker and d/o Thomas & Mary (Underwood) Cook of Wellsville, York Co PA. They married about Jun 1766.

Quaker Meeting Records 1681-1994 - Cane Creek Monthly Meeting, Alamance Co NC:
"Pg 51
At a monthly meeting held at cane creek 1st of 11th month 1766, The Representatives being called appeared - - - -
Mary Carson, formerly Cook, gave a paper of Condemnation to this meeting for going out in Marriage which was taken for Satisfaction - "

Uriah became a Quaker and in 1773 they moved to New Garden MM in Rowan Co now Guilford) NC, where he was received by request from Tom's Creek in Surry Co NC.

Encyclopedia of American Quaker Genealogy, by William Wade Hinshaw, Pg 530:
"CARSON.
1773, 5, 29. Uriah, of Tom's Creek, near the mountains, recrq."

Quaker Meeting Records 1681-1994, New Garden MM, Guilford Co NC, Pg 35:
"Sarah Carson Daughter of Uriah Carson & Mary his wife was born ye 16 of ye 2 mo 1766.
John their Son was born ye 7 of ye 12 mo 1768.
Ruth their Daughter was born ye 13 of ye 3 mo 1770.
Jacob their Son was born ye 6 of ye 1st mo 1775.
Uriah their Son was born ye 19 of ye 5 mo 1777."

Children with Mary:
1) Sarah Carson (1767-aft 1813); m. Byrum Ballard (1876-aft 1813).
2) John B Carson (1768-1840); m. Abigail Sumner (1762-1854).
3) Ruth Carson (b: 1770); m. John Harrold.
4) Jacob Carson (1772-1854); m. Martha Horton (1775-1855).
5) Bowater Carson (1775-1848); m. Charity Pierce (c1776-1852).
6) Uriah Carson Jr (1777-1860)

Notes: Quaker Old Style dates followed the Julian calendar with the first day of the year being March 25th and the year ending in February.
Quakers began using the Gregorian calendar in the year 1752. New Style dates conformed to our present day calendar, January to December.

Uriah's wife Mary is my 4x paternal Gr-Grandmother through her second marriage.