To stay accurate about spelling the name has several recorded spellings. Examples include Convery, Convey, Convoy, Convrey, Convry
1803 Newry Town Agricultural Census 1803 from
http://www.bagenalscastle.com/ancestryJohn Convery Drumcaselane (?) [Could be Drumcashllone or Drumalane?]
John Convery Domorey [could be Damolly??}
1823-36 irish Tithes applotment books
http://www.irishgenealogyhub.comCONVERY DOWN DONAGHMORE {this parish sits above Newry parish}
Convery, Edward Townland: Maddydumbreste Year: 1829
CONVERY ARMAGH KILLEVY (ORIOR UPPER) which includes Meigh
Convery, James - Townland: Clantigora Year: 1835
Convery, Peter - Townland: Killian Year: 1835
Convery, Widow - Townland: Clantigora Year: 1835
Convery, Widow - Townland: Killian Year: 1835
Griffiths valuation also helps in focusing locations , the quick list comes from
http://www.failteromhat.com/post1845.php Convery Edward Castle Street, Newry Newry Down
Convery Edward Hyde Market, Newry Newry Down
Convery James Bath Street, Ardglass Ardglass Down
Convery Michael Ballyvarley Aghaderg Down
Mc Convery James Drumbroneth Dromore Down and four in Belfast, any of which could be co ~down
However the irish times site show it’s a south derry name
Kingerswell generously gave you Meigh which is about 2 miles from Jonesborough [ie south Armagh.] Her parents were Bernard Convery and Bridget Murphy.
Ros Davies Co Down website has about 60 references to the name
http://freepages.genealogy.rootsweb.ancestry.com/~rosdavies/SURNAMES/C/ComCon.htm#conrhave you exhausted PRONI, ancestryireland, familysearch etc
JimG