It's a very complicated process and I'd need a chemistry degree to fully understand it, but our bodies do make glucose from fat. A triglyceride is basically three fatty acids bound together by a glycerol molecule. That glycerol molecule can be converted into glucose to provide energy to the brain.
As mentioned, people on a zero carb diet aren't void of glucose, and gluconeogenesis (by definition) also plays a role in converting substances other than carbohydrates into glucose, particularly protein. While I've seen figures that cite as much as 70% of undigested protein can be converted into glucose, that's not to assume that glucose necessarily enters your blood stream. Many studies have found strong evidence to support the concept that glucose converted from protein is a much slower process (perhaps 24 hrs or longer), and (aside from type 1 diabetics) rarely resulted in hyperglycemia. Furthermore, it seems that same glucose (originating from protein) was effective at addressing hypoglycemia.
The concept almost suggests that glucose from protein and carbohydrates has a similar relationship to basal (protein) and bolus(carbs) insulin in how they're each processed by the body. It's a loose comparison I know, but certainly one I find very interesting.