Is the story true about 'Umar ibn Al-Khattab discovering an incident of adultery, and 'Ali ibn Abi Talib warning him that he would be flogged if he mentioned the names of the perpetrators due to the absence of four witnesses? And is there a precise reference for it?

The incident mentioned was not found. It is probable that the questioner is referring to what was reported in the Sunan and other compilations: that Umar, may Allah be pleased with him, intended to stone a woman who gave birth after six months, whereupon Ali, may Allah be pleased with him, explained to him that she had an excuse, citing Allah's saying: ﴿وَالْوَالِدَاتُ يُرْضِعْنَ أَوْلَادَهُنَّ حَوْلَيْنِ كَامِلَيْنِ﴾ (And mothers shall suckle their children for two whole years) and His saying: ﴿وَحَمْلُهُ وَفِصَالُهُ ثَلَاثُونَ شَهْرًا﴾ (And his carrying and his weaning is thirty months). So, if weaning is twenty-four months, then gestation is six months. Thereupon, Umar let her go, and she later gave birth after six months. This account is also narrated about Uthman.