Well, generally speaking, a cold can cause discharge in the eyes. First, you should know that if you get cold, your immune system can be weaker. And it will make your eyes become fragile. And it is easy for some bacteria invade into your eyes. Your eyes will be infected. And then as we know that when you have eye infection, it eventually lead to discharge in your eyes. Also, eye discharge can be caused by allergies, bacteria, makeup, and chemicals. So I suggest that you schedule an appointment with your primary care physician. He or she can take your temperature, a history of your symptoms, and examine your eye's. If there is any question of whether the infection is bacterial, then you doctor will give you some advices. Of course, you can drink some medicine to get your cold recover first.