Yes, it can be a normal topic of discussion. It really depends on your intent and audience; you might tell your employer that you are ill if you need time off to rest. In this case, you usually don't need to get too specific (if it isn't a regular thing). If talking to customers, they typically prefer to hear that everything is great, no matter how you actually feel :) When talking to friends, feel free to give full details, if you feel comfortable doing so.
A more natural way of saying "I feel poorly" would be "I don't feel well", or more casually "I'm feeling ill".