He played hurling with his local club Redmonds and was a member of the Cork senior inter-county team for one season in 1892.
[citation needed] Conway first came to prominence on the inter-county scene with Cork as part of the Redmond's selection in 1892.
That year he lined out in his first provincial decider with All-Ireland champions Kerry providing the opposition.
The game was a controversial one as referee Dan Fraher changed his mind after initially awarding a goal to Cork.
Dublin, however, had walked off the field and, because of this, Cork were awarded the title.