What is the history of translation of Quran? Was there any political conflict due to it being translated or after it translated?

Medieval Western Christianity, early modern Catholicism (EMC), and medieval Islam cultivated an ideal of holy scripture in only one language (Latin or Arabic). However, the reasons for the two ideals were very different, and in all cases, the situation in practice was much more complicated.

In the West, the translation of the Bible into Latin was originally to bring it into the language of the people--hence the Latin Bible is called the Vulgate, as it was the "vulgar tongue," the vernacular. The evolution of Latin into the western vernaculars, and the rise of vernacular writing in the West, coincided with a period of time when the medieval Church sought to exert its spiritual and secular control over society, for the salvation of people and in a lot of cases the material benefit of clergy. One of the ways they did this was to, in theory, reserve Scripture to Latin. The idea was that educated, ordained clergy would read it and interpret it for people. Theologians argued that lay people interpreting Scripture would endanger their souls because they would inevitably get it wrong.

However, there are lots of places for holes in this. Friars and priests preached in vernaculars, including Scripture; there are actually quite a few vernacular Bibles floating around, and biblical material like collections of Gospel stories was very popular reading by the Reformation. So even when the medieval Church or EMC is advocating against vernacular Bibles, banning their publication, or burning Protestant/Anabaptist 'heretics', there is still vernacular biblical material floating around. And the concern is, ultimately, Church authority and lay people's perceived lack of intellectual/spiritual capacity to understand scripture for themselves.

Islam is different.

Islamic tradition holds that the Quran is the revelation of God to Muhammad: God speaks Arabic. Thus, the Quran cannot be translated if it is to be used for religious purposes (study and prayer). In the first centuries of Islam, scholars even exerted a significant effort to preserve secular poetry from the decades surrounding Muhammad, so they would have other examples of contemporary Arabic to study for insight into how to understand the Quran's language!

Because of the important of the Quran in Arabic as well as the intellectual context of early Islam in particular, a strong religious value developed on educating children (boys, but also girls) in reading and reciting classical Arabic. During the initial expansion of Islam out of Arabia, this was less of a problem in proselytization and other words I can't spell than you might think. The Umayyads were not that interested, generally, in mass conversion right away. The pattern we see over and over is that "Arabization preceded Islamization." That is, regions filtered Arab language and culture into their everyday societies even centuries before the bulk of the local population converted to Islam. Thus, some form of Arabic (perhaps not exactly the same as classical Arabic, but very similar in the Middle Ages) was already the language of most new converts in an area. And there were education institutions to teach them to read and recite it.

However, the idea that the Quran was "never" translated or "can't be" translated is a myth. A heavily entrenched one, but nevertheless. Although there are some other examples, the most studied case is Persia. Persia is one place where 'Arabization did not precede Islamization' entirely. Pre-Islamic Iran had a rich literary and scholarly culture that continued to exert its influence throughout the Islamic Middle Ages--even, in fact, influencing Islam in turn.

Abbasid scholars actually debated this question of whether the Quran could be translated. So not only do we have Persian Qurans, we have a significant amount of scholars putting forth the idea that this was religiously acceptable. In Persia, where Persian did persist as a spoken and written language, people converting to Islam could and would have been able to read the text in their own language. (Although it is possible that only Arabic was used in formal religious services.)

Further reading: Travis Zadeh recently (2012) published a really important book on medieval Quranic translation and theology that goes on to unravel the creation of the "untranslateable" myth. Unsurprisingly, it is not a story flattering to the western Middle Ages.

Travis Zadeh, The Vernacular Qur'an: Translation and the Rise of Persian Exegesis. Review/overview here.