Yes, of course it will make your eyes bloodshot. As we know that bloodshot eyes are common and usually occur due to eye irritation. So we can say that there are many reasons which can cause the eyes bloodshot such as the eye irritation, eye energy and lack of sleep. But for your situation, we can say that there is something to do with the high blood pressure which makes your eyes bloodshot. Because when you keep coughing, it will increase the blood pressure, and then it can affect your eyes. I suggest you should go and ask the eye doctor for the details
Yes, the eye bloodshot may be caused by long-time use of eyes in front of computers, high stress, and bad health and so on. If you have the coughing, the nerve system may be affected. The nerves in nose and throat may directly affect the eyes. And the immune system now becomes weak which makes the opportunity of getting eye bloodshot increase a lot.