I don't think it is normal. Although many people may say that the movement of bending over may cause the blood circulation turn over, the eyes still should not water. There must be some inflammation for your eyes. You'd better go to see the doctor and have an eye check.
Are you feel pain when you bend over? If you feel pain somewhere and eyes watering, it is normal. If not, you may suffers from watery eyes. You'd better see a doctor. Usually, watering eyes are usually caused by allergies and eye infections. If so, the doctor usually will prescribed your some medicine to help you. Good luck.
According to your description, you just bend over to pick something up, but led to the above bothersome results. The reason why this kind of symptom comes across, it can be explained by neural control. As we know that, when the facial nerves are subjected to external stimulation or oppression, you will feel muscle contraction. If it goes like this, nerve compression will make you fall into the dilemma you mentioned before.