At the time of the 1881 census she would have been only 6. The various computerised records would therefore calculate her year of birth to be 1881 - 6 = 1875. There is only 1 match on the Isle of Man for an Annie Quirk aged 6 in 1881, but she was born in Patrick...
Annie QUIRK, Female
Birthplace Patrick, Isle of Man
Head of Household William QUIRK
Dwelling: Treljah "Odd Fellows Arms"Public House
Census Place Patrick, Isle of Man
Family History Library Film 1342344
Public Records Office Ref RG11
Piece / Folio 5596 / 30
Page Number 13
However, in 1891 there are two 16 year olds listed. Both living in Patrick. One called Ann Quirk and the other called Annie Quirk. I'll look them up for you.