The phrase "fatal instinct" is found in the third paragraph of the short story "The Case for the Defence".
The story opens as the first-person narrator tells about a murder trial which they attended. The man accused of the murder of an old woman was "all but found with the body."
The phrase "fatal instinct" is used to describe the fact that Adams (the murderer) is caught with the weapon he used to murder the old woman. A neighbor (Mrs. Salmon), thinking she heard a noise at her own gate, sees Adams leaving the home of the woman he murdered. Adams feels a "fatal instinct" as he looks up to see Mrs. Salmon watching him.
Adams recognizes the fatal (deadly) instinct (intuition) that he has just made a mistake. Being seen by Mrs. Salmon puts him at the scene at the time of death.
Typically, fatal instinct is what keeps animals alive. The knowledge of something existing in such a way that could potentially end ones life. This is the same for Adams. He knows that Mrs. Salmon could be a witness which could out "a nail in his coffin."