The full 1861 entry is
Hobson Street, Oldham Below Town, Oldham, Lancashire
Ellen Garney Head M 45 Ireland
Ann daur 18 Factory Operative Ireland
JOhn son 13[5/8] Ireland (a bit of a scribbly date so not definite)
Patrick son 15 Ireland
Mary daur 5 Scholar Lancashire Oldham
RG9; Piece: 3009; Folio: 108; Page: 25
Let's follow them forward to check.
Did your John marry a Sarah?
1871 2 Court 2 House, Jackson Street, Oldham Below Town
John Garvey Head M 24 Cotton Spinner Ireland
Sarah Wife M 24 Card Room Operative Ireland
Thomas son 2 Lancashire Oldham
RG10; Piece: 4093; Folio: 122; Page: 39
also at 25 New Street, Oldham Below Town
Ellen Garvey Head Widow 46
Ann daur Unm 24 Throstle Spinner
Mary daur Unm 14 Twiner Piecer Lancashire Oldham
RG10; Piece: 4098; Folio: 142; Page: 2
Patrick is a little more difficult to spot, he doesn't appear to be in Oldham and there are quite a number of those of the same name around Lancashire, mostly just giving Ireland for their birthplace, any being more specific are living in unlikely places to be yours.
I don't see them in 1851 unfortunately, and judging by the ages and birthplaces of the children it may well be that they hadn't arrived in England at that time. I think your next step forward would be to get the birth certificate for Mary -
There are three options here
Mary Garvey - b 1844 Dec q Chorlton 20/198
Mary Garvey - b 1846 Jun q Manchester 20/761
Mary Ann Garvey - b 1845 Mar q Manchester 20/6_0 (I think it's 610)
If you ordered one of these on a guess and gave the criteria as mother's name Ellen costing £7, if you were wrong they would refund £4. In your position I would contact the GRO and ask them to do one of their three year searches - you state the three consecutive years - and ask them to check the Mary Garveys for one with mother Ellen, this should cost you £11.50 which might be a better bargain than guessing two wrong.
At least hopefully you should start out your search in Ireland with the names of both parents rather than just a surname.