When I have runny or stuffy nose, I get watery eyes also. Is there a connection?

Watery eyes. If you have an upper respiratory infection such as a virus, it can effect the nose and eyes. Watery eyes (like crying) can also lead to a stuffy nose since the tears drain into the nose through the naso-lacrimal drainage system.
Yes. They can be related. Tears drain from the eyes into the corner of the eyelids and then into the nose. If you have a stuffed nose, the tear drain can be squeezed closed and tears will not have anywhere to go. They will ultimately collect on the eyelids and run down your face.