Is it possible to have bright red bleeding and still be pregnant in the first month?

Yes, it is possible: Bleeding in early pregnancy is unfortunately not uncommon and most often is of unclear cause. It is often attributed to "implantation" or to "placentation issues" and resolves spontaneously. A combination of your other symptoms, your betaHCG level and your pelvic sonogram findings may provide you and your doctor a sense of direction. Consult your gynecologist.