Can going out in the cold make baby sick?

My daughter is 6 months and we live in Phoenix, AZ. We go out a lot to walk, she wears warm clothes but it' still little chilly out, my question is can the cold weather make you sick? I ask this because she started coughing and my mother said maybe it's because I give her bath every night before bed and that we walk out, my house is not cold, the temp. stays around 65 because I hate sleeping when it's very warm. My baby wears a little wearable blanket and stays warm. I am really confused, is it possible the coughing came from our walks and the nighttime baths? She loves her bath and it helps her go to sleep.. Also her coughing is not bad but it's unusual. What am I doing wrong and is my mom right? Please help. Another thing, I recently started giving her organic rice cereal and other than that she gets breast milk all the time, she only eats the cereal for breakfast and dinner and I breastfeed her after she eats her cereal.