If you mean the ancestors of this couple, Leigh M. Shaw (b. Fredericton Junction ca. 1914) was the adopted son of Bernard Shaw. Leigh married Berniece in Peel Parish, Carleton Co., NB. Bernard adopted Leigh from an orphanage.
I am having difficulty identifying Bernard Shaw. The name may be incorrect. I see two Bernard's, both too young to have adopted Leigh. Possibly he was the B. B. Shaw b. 2DEC1843 with a wife, Lillie (12DEC1850) in Brighton Parish, Carleton Co., NB in the 1901 census. He was B. B. Shaw age 47 in Brighton Parish in 1891 with a wife, Millie J. (41.) He seems to be 'Beali' Shaw (37) with a wife, Lillie (30) in 1881.In 1871 he is Bela B. Shaw (27) living with likely a brother, Morris Shaw (25.) According to his headstone, Bela B. Shaw was b. 2DEC1843 and d. 8JAN1919. He and his wife, Lily Jane (1850-1930) are buried together in Carlisle Cemetery, Carleton Co., NB. Bela B. Shaw married 'Lillia' J. Wark at her father's home in Andover, Victoria Co., NB OCT1872.
There was a Bernard Shaw (ca.1844/5-27SEP1868) in Carleton Co., NB but he died before Leigh was born. It also appears that he actually died 27SEP1867 of small pox on board a ship, "Glimpse", on its way from San Francisco to the Fraser River in what is now Vancouver, British Columbia, probably a gold rush at the time. So he likely isn't even buried in NB and the headstone was likely raised long after his death, thus the error in the year-of-death.
There appear to be three problems. The name of Leigh's adoptive father, as given on his marriage record, seems to be incorrect. There is no evidence of a Bernard Shaw of an age to have been that father. The Shaw family is very large; perhaps several families, making it necessary for an extensive examination to come to a solution.
Fredericton Junction, in spite of it's name, is in Blissville Parish, Sunbury Co., NB. It is near Fredericton, but is a different place. I do not see any Shaw's in Sunbury Co. in, for example, the 1911 census. The possibility that Leigh was born a Shaw, and related to his adoptive family, seems remote.
Census records go to 1911 and available vital statistics to about 1915. This is to protect the privacy of living persons. It is possible there is a birth record for Leigh M. Shaw, but his birth surname would have to be known.
Berniece Ethel Kinney (b. Florenceville, Carleton Co., NB ca. 1920) was the daughter of George Atkinson and Mary Elizabeth (Howard) Kinney.
George Atkinson Kinney was b. Marysville, St. Mary's Parish, York Co., NB. I think he was the George Kinney b. 29MAR1856, in Peel Parish, Carleton co., NB in the 1901 census. His parents at the time were Asa Kinney (18FEB1834) and Martha (25SEP1829.) His marriage record states that he married Mabel E. Hayward in Carleton Co., NB 13MAR1905. In this case he is given as born in Bristol, Carleton Co., NB ca. 1867. He was a bachelor and his parents were Asa and Martha Kinney. This sort of discrepancy in birth-dates is normal.
Mabel E. Hayward was b. England ca. 1882. She was a widow, and the d/o (daughter of) Alfred and Sarah Hayward. I think she was the Mary Elizabeth Howard above, which shows the reliability of these records. According to George's death records, he was b. in York Co., NB (possibly Marysville again) 22OCT1864. (Yet another birth-date.) His father was 'Acer' Kinney and his mother was Martha Kinney. He dies in the Fisher Hospital in Woodstock, Carleton Co., NB 6MAY1936. In the birth of one daughter, Henrietta Phyllis Kinney, he is George 'Ackerson' Kinney and his wife is Mabel Annie Elizabeth Howard. With minor variations (i.e, Mabel Elizabeth Annie Howard) her maiden name is usually Howard, so that is likely the correct version.
Mabel appears to be the Mabel Howard b. NB 27AUG1881, working for G. W. and Phoebe VanWart in Woodstock in 1901. That she was born in NB may be inaccurate.
George A. Kinney (1864-1936) is buried in the East Florenceville United Baptist Cemetery. Mabel (1880-1976) is buried in the East Florenceville Anglican Church Cemetery.
Birth records also give Asa Kinney's wife as Martha King. The marriage record of a son, also Asa, gives them as Andrew and Martha Kinney, again illustrating the uncertainty of these records. Asa Robinson (or Robertson) Kinney is given as b. Bristol, Carleton Co., NB 22JUL1856, and d. Bristol 14JUN1938. His father was 'Ostin' (Austin) Kinney and his mother was Temperance Dyer. In Kent Parish (wher Bristol is), Carleton Co., NB in 1851, Austin 'Kenny' (b. ca. 1829) was the s/o Asa (ca. 1784) and Sarah (ca. 1800) 'Kenny'. Temperance Dyer (ca. 1832) lived in the same parish with her parents, George (ca. 1800) and Mary (ca. 1805) Dyer.
I strongly suspect that Asa Kinney was a member of a family from New England that settled first in Maugerville, Sunbury Co., NB ca. 1763; of a group of families called 'pre-Loyalists' New Brunswick was still part of Nova Scotia and very little settled at the time. When the Loyalists arrived in 1783 many went up the River Saint John to what would become York and Carleton Counties. Asa Kinney was certainly in Sunbury Co. in 1818. This seems to be the case, and he seems to be the Asa Kinney, s/o Israel and Susannah (Hood) Kinney at:
While this Asa is born about the right time and does go to Bristol, Kent Parish (not Kent Co.), Carleton Co., NB, his wife is given as Elizabeth Tompkins. Elizabeth died in 1842, so Sarah, in 1851, was Asa's 2nd wife and Austin, while not listed in the above compilation, is likely Elizabeth Tompkin's son as well. The Tompkin's I would suspect to have been Loyalists from Westchester Co., New York. See also:
where the Kinney's are said to have been involved in the Salem witch trials:
At this point you need the advice of someone more familiar with the Kinney's in Carleton Co.