Lets make one thing clear..... i think you mean to say you see blood in saliva, not sputum. Because sputum is some thick dirty greenish substance which comes out of your mouth when you cough. While brushing, if there is infection in your gums, due to friction of gums with your (hard) toothbrush, the infected gums may rupture and bleed, which you can see in the saliva mixed with toothpaste foam that you spit.

Hello, bleeding could be because of deposits on your teeth or food lodged in your teeth. Visit a dentist get your scaling and polishing done. Brush twice daily, maintain proper brushing technique, floss regularly and rinse after every meal. Visit dentist every 6 months.

Hello there.  I can understand your concern.  Does the blood come only in the morning while brishing the teeth or is it present all through the day? If its only while brushing the teeth, then its from teeth itself. So do not say that. If you are still scared of cancer, get it checked out by a pulmonologist in person for further evaluation and management. Best regards. Dr. Anil kumar jain. Bangalore.