Hi @Pyrre, possibly, yes. Time for some quick maths:
Let's take an average chap that weighs 70kg. He mixes 350ml of ice cold water (2°C) with Jake (20°C) and gets a shake of 500g (8°C). He drinks the shake and, doing so, becomes a guy of 70.5kg whilst the average temperature of his body drops 0.21°C to 36.79°C.
As (500g x 8°C + 70,000g x 37 °C) / 70,500g = 36.79°C [-0.21°C]
To heat itself back up to 37°C, his body needs energy. Conveniently, the unit of energy is the calorie, and a calorie is defined as the amount of energy needed to raise the temperature of 1g of water by 1°C. So to warm up his 70,500g of body mass with 0.21°C, he needs 15 kcal.
As 0.21 x 70,500 = 14,805 calories = ~15 kcal.
That's it then .
I have to disappoint though. Aww. This wasn't really as accurate as it looked. Your body produces a lot of heat doing its usual stuff, so depending on e.g. the temperature, your BMR and activity, your body might actually be working to cool itself off. In which case drinking cold water can even save energy.