No. Rabbits are induced ovulators, meaning that the female's ovaries only release eggs when triggered, and the trigger is mating. Human females release one egg every month, and if it isn't fertilized the body "flushes the system," which is our period.
Blood on the genitals or in the urine means there is a serious health problem, such as a urinary tract infection, pyometra (uterine infection), uterine cancer, birthing problems, etc.