Did Omar Ibn Al-Khattab imprison his son Abdullah, Abu Hurairah, and a third Companion due to their companionship with the Noble Hadith, claiming that this would distract them from the Holy Quran?

There is no evidence that 'Umar imprisoned anyone for writing Hadith. However, he was strict with those who narrated Hadith, and he demanded proof from Abu Musa al-Ash'ari when he narrated the Hadith of seeking permission. 'Umar intended to write down the Sunan, but he later abandoned the idea for fear that people would forsake the Qur'an. The matter of writing down Hadith and codifying the Sunnah was settled after 'Umar ibn 'Abd al-'Aziz ordered it, fearing the fading of knowledge and the departure of scholars.