John James Quine married Margaret Cowin on May 28, 1864. Their son James John was christened November 13, 1864.
James Quine son of Margaret Quine was christened three months later on February 5 1865. Presumably a different mother.
It is possible that his father's name was used in the 1871 census.

Note that another James Quine appears in the IOM census data. He is also an apprentice baker.
1861 Onchan village
Margaret Corin (or Cowan?) 77 widow b. Onchan
Elizabeth Creer Unm 34 washerwoman b. Onchan
George Stafford 10 unm son
George Quine 4. son

1871 no. 31 Village Onchan
Elizabeth Creer Unm 46 laundress b. Onchan
George Stafford 20 unm son Ag. lab.
Robert Quine 14 unm son b. Onchan
James Quine 9 unm son b. Onchan
1881 Village Onchan Entry 120
Elizabeth Creer Unm 53 laundress b. Onchan
George Stafford 20 unm son Ag. lab.
James Quine 19 unm son b. Onchan Baker's apprentice
Robert Quine married 23 Ag. lab
Annie Quine wife 27