Is there a Hadith that encourages reading the Qur’an between Fajr and Duha, and what is its reward?

There is no reported hadith (definition) that encourages recitation after Fajr, but there is an authentic hadith that encourages dhikr (remembrance of Allah) after Fajr in the mosque until sunrise. This is the saying of the Prophet, peace and blessings be upon him: "Whoever prays Fajr in congregation, then sits remembering Allah Almighty until the sun rises, then prays two rak'ahs, will have a reward like that of a complete Hajj and 'Umrah (definition)." The Qur'an is the best form of dhikr, so a Muslim can complete the prescribed morning adhkar and then engage in Qur'an recitation.