We heard from some of Muslim scholars saying that the big day (Qiyamat) will occur on the Friday and that will be the 10th of Muharram. Is there any information in this regard in Qur'an and Hadith?
All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
Al-Tirmizi, Abu Dawood, Al-Nasa'e and Ahmad reported in a good Hadith from Abu Hurairah (Radiya Allahu Anhu) that the Prophet said: "The sun has not shone in a day better than Friday. It is the day in which Aadam, may Allah exalt his mention, was created and was descended (to earth) and was forgiven and died. It is the day in which The Hour (Day of Judgement) will take place". So, this is an evidence that the Day of Judgement will take place on this day (i.e. Friday). As for the claim that it will be the 10th of Muharram, we do not know any evidence for that.
However, we draw your attention that what the person needs to know more is what good deeds he/she has for that day. Al-Bukhari and Muslim reported from Anas that a man asked the Prophet about the Hour saying: 'When will the Hour be?' The Prophet said: "What have you prepared for it?" The man said: 'Nothing except that I love Allah and his apostle'. The Prophet said: "You will be with those whom you love.”
Allah Knows best.