Conners had four bogeys - on the second, third, sixth and twevelth holes - and is tied for 114th.
It's just the second major tournament for the 25-year-old who is playing on the Web.com tour this season. Conners played in the Masters in 2015.
He is 33rd on the Web.com Tour money list and has two top ten finishes.
The only other Canadian at the US Open, Adam Hadwin, finished at four-under 68 and is currently tied for 7th.
American Rickie Fowler is alone atop the leaderboard at 7-under 65.