This might or might not be serious. The likelihood of serious or not depends on your age. For example, a non-smoker less than age 30, almost certainly benign vs a smoker over age 60 having a reasonable chance of the bleeding coming from a tumor. Seriousness would depend on the source of the blood and could range anywhere from a slow nose bleed into the throat to a lung disease that is bleeding. Examination of your nose and throat and maybe a CT of sinuses would be a good start, along with a chest x-ray. You may have to have a direct examination of your throat and airways to find the source. This is done with what is called a fiberoptic bronchoscope.
You should persist with the evaluation until one or another doctor can tell you the cause of the bleeding with certainty.
Could be a tumor or perhaps TB (rare). See your physician.