I'd get her to a dentist first of all. Is there blood when she spits out brushing her teeth? Could be gum disease? The gp will probably ask if shes seen a dentist before anything else.
She could also as someone else has mentioned chew her cheeks, im 29 and ive done this for years! I don't experience any blood though so i can't do it that badly.
She could also be about to lose a tooth? teeth grinding would probably make a loose tooth bleed more as well. See about getting her a retainer if you can. I know how bad grinding can be, myself and my kids do it and my son ripped 2 half teeth out and my girl and i have worn down the enamel on ours. It's best prevented if possible.
gums more than likely. I don't know of many 6 years olds that brush properly and often. :)
Also could be like a tiny nose bleed that passes from the nasal cavity down into her throat.
I wonder if she's chewing on the inside of her cheeks enough to make it bleed. She should definitely go to a doctor.
Sounds like a teeth problem ,may be a good idea to have her teeth looked at ,at the dentist ..