These two matches have the record for a team:
Round 12 1945
Norwood 0.11 3.15 6.17 6.22 (58)
Glenelg 0.4 1.7 3.12 5.13 (43)
Glenelg, 1500
Round 17 1992
Eagles 0.7 7.9 13.12 13.21 (99)
Central 3.5 8.5 10.5 13.8 (86)
Elizabeth, 2055
Scorers:
Eagles: Neave 3.2, Niemann 2.8, Taylor 2.2, Morphett, Pyman 1.1, Haylock, Phillipou, S.Sziller, Totham 1.0, Klemm, Pesch 0.2, Rushed 0.3.
Central: Coffee 4.4, Tongerie 3.1, Smith 1.2, Dixon, Girdham, Lounder, Sutton, van Dommele 1.0, Abbott 0.1.
As I have highlighted, each occasion saw 16 behinds across the two quarters when each team went goalless in them.
For a team to go goalless in three quarters, the record for that is 13 behinds:
Round 10 1954
North 3.2 3.7 6.11 8.11 (59)
Sturt 3.4 3.9 3.14 3.17 (35)
Naracoorte, 4500
Round 8 1977
West 8.5 13.10 17.15 21.22 (148)
Woodville 0.1 0.7 0.13 4.18 (42)
Richmond, 2348
For one quarter, it's also 13:
Round 16 1937
South 3.5 4.10 4.23 7.25 (67)
Torrens 1.1 3.5 5.7 9.10 (64)
Thebarton, 5000
And Port went goalless in the 2nd Semi of 1906, whilst scoring 12 behinds:
2nd Semi 1906
North 2.0 2.0 4.2 5.4 (34)
Port 0.4 0.8 0.9 0.12 (12)
Adelaide
For the 1937 match, the article on it can be seen here. In this day and age of "record any stat you can think of", it's notable that they documented who hit the goalpost at that time.
http://newspapers.nla.gov.au/ndp/del/pa ... oomLevel=1
That match was also one of three matches that the current oldest living League player played in. George Kersley turned 100 this year, and in the "Best of Both Worlds" book, can be seen photographed with Malcolm Blight, Lindsay Head, Fred Bills and Ray Huppatz; players he coached in the junior grades, along with Bob Hank.