The closest in the 1881 census (on FamilySearch) to what you have written is:
James QUIRK Head M Male 67 Rushen, Isle of Man, Mariner
Annie J. QUIRK Grand Daughter Female 3 Rushen, Isle of Man
Annie YATE Daur M Female 21 Rushen, Isle of Man, Mariners Daughter
William F. YATE Grandson Male 4 m Rushen, Isle of Man
Source Information:
Census Place Rushen, Isle of Man
Family History Library Film 1342348
Public Records Office Reference RG11
Piece / Folio 5609 / 6, Page Number 6

I've had a look at the original scan, and it could be either a "T" Tate, or a "Y", and also Annie I., not Annie J.

IGI has Ann QUIRK & EDWARD TATE Marriage: 21 AUG 1880 Kirk Rushen.

Brian Lawson's birth regts have:
Name Date Par Page Father Mother
TATE, Gertrude, [born] 12 Nov 1882 Rus 84 [father] TATE, Edward [mother] QUIRK, Annie.
His burials have:
TATE, Gertrude 5 wks Rus 6 Jan 1883

Births:
Name Date Par Page Father Mother
TATE, William James 12 Nov 1880 Mal 32 TATE, Edward QUIRK, Ann

You could order the birth certs and the marriage record using these refs.

Sue