Only if you get hypothermia (a specific medical condition that results from extreme temperatures lowering the
core body temperatures ~ which is not an easy thing to do since our bodies have an amazing temperature control system especially when it comes to core body temperatures).
The common cold is caused by a
virus. It is caught and spread by the transfer of virus particles. What can make you more prone to illness, is extreme tiring or weakening of your body's immune system (usually caused by medications such as transplant patients, people on chemotherapy, those with auto-immune disease, or other problems such as HIV/AIDS) which makes it harder, if not impossible, to fight infection.
Humans do lose heat rapidly through their head, which is a process that would accelerate with wet hair - usually when you get chilled to the point of shivering, the process the body uses first to help warm itself, you will be uncomfortable, but that itself is a cure, not a cause of lowered temperatures. If you get to the point of shivering and then having the shivering stop (which happens when you are at or near hypothermia), you may have a lowering of your core body temperature which may affect your immune system and therefore you could be more likely to catch a cold. This is a rare event. So, having wet hair and being in cold air wont make you sick unless you get to the point of hypothermia.