I had from some people that every verse of the Qur'an has a particular event that happened before or after Allah said those words like wise Hadith too.
So is this true, and can I ask the narration of any verse and Hadith on this site
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Many passages in the Quran have events that happened to prompt Allah revealing that passage.
An obvious example which you don't even need a narration to tell you about:
They ask you, [O Muhammad], about the new moons. Say, "They are measurements of time for the people and for Hajj." And it is not righteousness to enter houses from the back, but righteousness is [in] one who fears Allah. And enter houses from their doors. And fear Allah that you may succeed. (2:189)
Here, clearly, what happened to prompt this passage is someone asking the Prophet about the new moons. It is literally stated.
An example where it isn't as obvious:
Say, "O disbelievers, I do not worship what you worship.Nor are you worshipers of what I worship. [...] (109)
This verse was revealed when the people of Makkah suggested to the Prophet (SAW) that they all worship the idols for one year and then worship Allah for one year as a compromise. The verse is a direct response to this statement.
Knowing these narrations and events that prompted the revelation of verses is the field of Asbab An-Nuzul (Reasons for Revelation). You can read tafasir to know the reason for many verses, and you can also read books dedicated to Asbab An-Nuzul like the book by Al-Wahidi (here).
However, there are a large amount of verses whose reason for revelation may be unknown or may not even exist in the first place.
For example, Allah says:
By the sun and its brightness And [by] the moon when it follows it And [by] the day when it displays it [...] He has succeeded who purifies it (his soul), And he has failed who instills it [with corruption]. [...] (91)
This Surah is a general warning to people about the consequences of not following guidance. There isn't a specific event that prompted its revelation. Allah revealed it to warn people, and get across the message.
The conclusion is that there are many verses and passages that have events corresponding to them. Likewise many verses do not have any events corresponding to them. You should read tafasir books to learn if specific verses have events attached to them.
As for hadith, many hadith already mention the context. If not, sometimes you can find different narrations of the same hadith that mention the context.
However, it is also possible there is no context to be found except the Prophet (SAW) advising people. For example, the Prophet (SAW) said:
Islam is based on (the following) five (principles):
To testify that none has the right to be worshipped but Allah and Muhammad is Allah's Messenger (ﷺ).
To offer the (compulsory congregational) prayers dutifully and perfectly.
To pay Zakat (i.e. obligatory charity) .
To perform Hajj. (i.e. Pilgrimage to Mecca)
To observe fast during the month of Ramadan. (Sahih Bukhari)
Knowing the context of this hadith isn't really that important. It is simply the Prophet (SAW) teaching people what they need to know.