Why cold water results in cough
I am no expert in this but I had asked my doctor the same question the last time I fell sick and I'll try to paraphrase his explanation. 
"There is this protective lining on our respiratory tract called "Respiratory Mucosa" that acts as the first line of defense against pathogens. But when you consume cold food and beverages , this membrane gets congested or shrinks such that it exposes the respiratory tract to pathogens . This is more prevalent if your body has an elevated temperature such as after spending a lot time outdoors during summer or after a workout. Cold food as such doesn't make you sick but they make it easier for the pathogens."
A quick Google search turned up this
http://www.thehealthsite.com/diseases-conditions/why-does-drinking-chilled-beverages-cause-sore-throat/#
Your cough could be a symptom of having a sore throat from infection.