A cold is caused by a virus and you need to catch it from someone else who has a cold. So if the window was open and someone with a cold walked past sneezing or coughing then possibly. The only effect otherwise is that by sleeping under the window, it created the right conditions for a cold he had already caught to develop. If you do sleep with a window open and it is cold, you can have the symptoms of catching a cold (blocked nose, dry throat, thickly cough etc.), but this will soon pass once you have warmed up and had hot food and drink as they are only the bodies natural reaction to being cold.
Definitely you can catch a cold if you sleep under a window on a cold night. This is because the most of the people lack a metabolism through which they can fight cold and related infections. And sleeping in a cold atmosphere triggers cold infection so yes your husband has caught this cold by sleeping under the window.