Can turkey stock be left out overnight?

No.

According to the United States Department of Agriculture (USDA), cooked turkey stock should be discarded if left out at room temperature for more than 2 hours. This is because bacteria can grow rapidly at room temperature, and eating food that has been contaminated with bacteria can cause foodborne illness. Refrigerate your cooked turkey stock in a covered container within two hours of cooking. It should be used within 3 to 4 days.