Queer (Adj): differing in some odd way from what is usual or normal; of a questionable nature or character; suspicious; shady

From A Jury Of Her Peers: ” He didn’t begin at once, and she had noticed that he looked queer–as if standing in that kitchen and having to tell what he had seen there yesterday morning made him almost sick.”

From what I finally understand from this passge, is that, she was noticing that Mr. Hale looked odd, in which, she was trying to figure out his questionable nature that made him sick.