Who of the two embraced Islam first: Omar or Hamza, along with a brief account of that?

Hamza, may Allah be pleased with him, embraced Islam three days before Umar, towards the end of the sixth year of prophethood, most likely in Dhul-Hijjah of that year.

The reason for Hamza's conversion was that Abu Jahl had insulted the Messenger of Allah, peace and blessings be upon him, near Safa. A freedwoman of Abdullah ibn Jud'an witnessed this and informed Hamza when he returned from hunting. Hamza became enraged, entered the Mosque, and struck Abu Jahl with his bow, saying to him, "O you who stains his buttocks yellow, you insult my nephew while I am upon his religion?" Thus, Quraish realized that the Messenger of Allah, peace and blessings be upon him, had gained strength.