No, you are not releasing an egg during your period.
Understanding the Menstrual Cycle and Ovulation
The menstrual cycle and the release of an egg (ovulation) are distinct phases. Your period, or menstruation, is the shedding of the uterine lining if conception hasn't occurred. Ovulation, on the other hand, is the release of a mature egg from an ovary. This typically happens around day 14 of a 28-day cycle, about two weeks before the start of your period. A surge in luteinizing hormone (LH) triggers this release. Source: Ovulation: Calculating, Timeline, Pain & Other Symptoms, Source: Ovulation and fertility - Better Health Channel
While you might be able to conceive during your period, this is because sperm can survive in the female reproductive tract for several days, potentially fertilizing an egg released later in the cycle. Source: Could I Get Pregnant if I Have Sex During My Period? (for Teens ...] The period itself doesn't involve the release of an egg.
During your period, follicles in your ovaries are stimulated to grow, each containing an egg. However, only one egg is typically released during ovulation. Source: What is Menstruation? | Get Facts About Having Your Period, Source: Women, How Good Are Your Eggs? > News > Yale Medicine]
In cases of anovulation (absence of ovulation), a woman may not release an egg regularly or at all. This can lead to irregular periods or the absence of periods altogether. Source: Anovulation: Signs, Symptoms, Causes & Treatment
In summary: Menstruation and ovulation are separate processes. You are not releasing eggs during your menstrual period.