All our body's cells can use glucose, and a few are dependent on it. People consuming an average diet get plenty of glucose from their diet. Starches (plentiful in grains including flour, potatoes, etc) are essentially long chains of glucose. In addition naturally-occuring sugars as add as added sugars are plentiful in the diets of most people. However, if carbohydrate is not being consumed, the body will make glucose from other sources.
The process of gluconeogenesis takes place primarily in the liver, where glucose is made from amino acids (protein), glycerol (the backbone of triglycerides, the primary fat storage molecule), and glucose metabolism intermediaries like lactate and pyruvate.
Source: Dietary Reference Intakes for Energy, Carbohydrate, Fiber, Fat, Fatty Acids, Cholesterol, Protein, and Amino Acids (Macronutrients) (2005), Institute of Medicine, Food and Nutrition Board, National Academy of Sciences.