I completed the quest “A Deadly Plot,” it shows up in the completed section. I even Played Gwent and got the Gwent card to complete that quest. I then work on "The Isle of Mists" quest. I get transported through the fog in the “The Isle of Mists” quest, and the screen flashes “Quest Failed: A Deadly Plot.” The quest is also moved from the “completed” section to “failed” section. Can anyone help? Even if it's console commands.

Do you have any mods installed? Specifically, do you have a fast travel mod?

Nope, no mods.

I've heard of this happening before, with the culprit being a fast travel mod, that's why I asked. Since you say you have no mods, I would ask over at the CD Projekt forums in case it's a known bug.