Yes cold weather does suppress and weaken your immune system so you're more susceptible to colds and flu viruses when you're in the cold weather.
The cold weather doesn't actually make you sick or cause the cold or flu virus because you need the flu or cold virus to get inside your body to actually cause the illness.
But when you're cold your body cannot fight the flu virus or cold virus off as good as it would if you were not cold.
You can also catch pneumonia more easily when being cold such as in a cold house etc.
So try to keep your house temperature warm enough to prevent sickness as you may be saving money on heating fuel but it could cost you your life or a medical bill from the hospital.
I always keep my house temperature in the winter at 75 F to keep myself and my family warm enough.
If the temperature in the house gets to 65 F we're feeling too cold.