The most common cause of bleeding from gum next to the teeth is accumulation of plaque or accumulation of hardened plaque called Calulus(tartar). The bacterial toxins from these inflame the gum in contact and later infect it causing bleeding either spontaneously or while brushing. Pain may not be there, leading to the patient neglecting it.
Next Steps
Go to a dentist evaluate, get the teeth cleaned, observe correct brushing and flossing. You should be relieved of the symptoms within a week to 10 days.
Health Tips
Unless it has progressed to periodontal disease or you have some rare systemic condition.