No, 4 calories is what I use, but protein has around 5 to 5.5 calories per gram in it, but because it takes around 20-25% of those calories to digest, we count it as 4 calories per gram. So yes protein burns calories to digest, but that is already calculated in the calorie count.
The amount of calories in a gram of carbohydrates varies a lot too, it isn't just 4, it's more like 3-4.5 depending on the carbohyrate.
Plus this all assumes that you are digesting most of it, if for some reason you can't digest it well enough, the numbers can be lower.
Protein is closer to 3 calories per gram. This is from Martin Berkhans book, The Leangains Method,
DIT: Protein
"Protein leads the way in body weight regulation, producing the greatest satiety and highest diet-induced thermogenesis of any macronutrient: a staggering 20 to 35 percent.56 It’s a large enough number that any reasonable person might propose counting protein grams as 3 calories instead of 4. That’s exactly what Dr. Geoffrey Livesey, one of the world’s leading authorities on metabolism and human nutrition, did during the Food and Agriculture Organization (FAO) meeting of the United Nations in 2001.57 There, experts on human nutrition gathered to discuss and improve our understanding of dietary issues, including the energy content of food. We need to take into account all the considerable knowledge we’ve learned since 1889 and start applying it. —Dr. Geoffrey Livesey Livesey suggested the prevailing and outdated concept of metabolizable energy be replaced by net metabolizable energy, a model superior in every way that—among other things—accounts for DIT. While there were no academic objections to Dr Livesey’s proposal, the administrative headache of pushing this through must’ve seemed daunting. After deliberating for two years, the FAO decided to stay with ME. (ref. 11) Why didn’t they take strides to publicly update and recalculate calories per the evidence of current science? Cost to the food industry of changing labels. The FAO was attempting to harmonize regulations across agencies worldwide. Introducing NME and replacing the latter would have slowed, or even dismantled, that process.58 The delay ultimately came down to expenditure, timing, and special interests—an unfortunate trifecta. No surprise—but an impetus, I suppose, for this book. Why is the DIT of protein so high? In short, protein contains nitrogen, which needs to be removed before amino acids can be put to use. Once amino acids are freed, they’re used for energy-demanding processes such as protein synthesis and de novo gluconeogenesis (DNG). Studies looking at the latter found 20 to 33 percent of calories from ingested protein were lost, depending on the nutritional state of subjects—20 percent when fed, and 33 percent when glycogen depleted (ref. 1,12). Indeed, the two biggest factors that affect the DIT of protein are protein synthesis and DNG. Initially, protein synthesis is responsible for two-thirds of the increase in energy expenditure. DNG largely makes up the other third. With time, however, this shifts into a more even split, as protein synthesis cannot remain elevated indefinitely. While there are several minor contributors to the DIT of protein, it’s fair to say it’s mainly caused by protein synthesis and DNG (ref. 12,26). It’s also worth mentioning that the DIT of protein differs slightly depending on food source. High-quality sources like meat, eggs, or dairy provide a different DIT compared to lower-quality protein (e.g., vegetables and grain products). In this context, the amino acid profile of the aforementioned foods determines the protein quality. Meat contains an abundance of essential amino acids, making it high quality, while proteins in vegetable and grain products do not. High-quality proteins therefore yield a higher DIT, but this is of negligible consequence in a diet containing a mix of protein sources. Flank steak is the king of the grill, but can also be enjoyed on a frying pan like any other meat. Just make sure the cut is not too thick. Since the DIT of protein varies depending on the physicality and habitual diet of the consumer, it’s not far-fetched to think weight training, which affects protein synthesis positively, also affects DIT. But the extent thereof remains speculative until research proves otherwise.59 Along with being the most thermogenic nutrient, protein is also the most satiating (ref. 1). While attempts to draw a direct link between satiety and DIT, implying the former causes the latter, have been made in theory, they’ve failed in practice (ref. 13,14). Indeed, the “filling” properties of protein are largely explained by its impact on peptide YY (PYY), an appetite-regulating hormone (ref. 13,14,15)."