Probably not. If you and your child are appropriately dressed and not too cold at 65 degrees then there is no problem. Cold air doesn't predispose to "colds" and cold air induced asthma usually requires lower temperatures.
Cool house & asthma. Cold air can sometimes trigger asthma attacks, but this is usually related to a quick change in environment. Heat that is too high can irritate an asthmatic's lungs, as can air that is too dry. A steady cool and not too dry temperature should be fine, but it is always best to take your cues from the child!