Is it permissible to complete fasting during menstruation, knowing that one believes in the Quran and fasts secretly, and is unable to declare Islam publicly?

You must utter the two declarations of faith (Shahadatayn) to embrace Islam, even if it is secretly, to yourself, if you fear for your life and your religion. You can declare it publicly later when you are able.

As for fasting and prayer for a menstruating woman, there is no fasting or prayer during the days of menstruation. Fasting must be made up after menstruation ends, but prayer does not need to be made up. This is agreed upon by all Muslim scholars.

Regarding the rest of the religious rituals, perform what you are able to, and what you are unable to perform due to fear for yourself or your religion is waived. However, you are obligated to migrate if you cannot observe the rituals of religion in your country and are capable of doing so.