When does a Muslim pray the Fajr prayer if he missed it, knowing that prayer between Fajr and Dhuhr is forbidden?

Your statement that prayer between Fajr and Dhuhr is forbidden is incorrect. Rather, what is well-known is the dislike (karahah) of supererogatory prayers (nafilah) that have no specific reason, from after Fajr until the sun rises, and when it is at its zenith in the middle of the sky until it declines. Obligatory prayers are not included in this prohibition, due to the Prophet's (peace be upon him) saying: "Whoever forgets a prayer or sleeps through it, its expiation is to pray it when he remembers it." And Allah knows best.