Not everyone gets diarrhea when they have a cold virus but it can happen to some people and having diarrhea with a cold virus can be normal for some people.
If you have a cold virus and never experienced diarrhea with the cold or flu then the diarrhea could mean something more serious is going on.
So if the diarrhea lasts longer than a day or two I would recommend going to urgent care or the emergency room to get checked out.
Diarrhea can occur normally in some people with a cold or flu but normally diarrhea or vomiting doesn't usually happen when you have a cold or flu.
But if you have a stomach ache along with the cold or flu then diarrhea can happen.