Yes your nose can bleed when the weather turns cold and you come into contact with the cold air.
Mostly the cold air when it's cold outside during winter is dry air and that dry air dries out your nasal passages and causes your tiny blood vessels in your nose to crack and then bleed.
The nosebleeds will go away on their own and are nothing to worry about.
You can help prevent your nose from bleeding in the winter and cold air by using a humidifier in your home or bedroom when you sleep which helps to add moisture to the air and your nose.
You can also put some Vaseline or petroleum jelly inside your nostrils to help keep your nose moist and prevent them from bleeding due to them drying out.
Getting nosebleeds in the winter is a normal thing and it happens to me too.
I also used to get nosebleeds when in my house during the winter due to the dry air from heater being on.
I had a humidifier installed on the furnace which helps constantly add moisture to the air coming out of the furnace vents and I stopped having nosebleeds.