Have there been controversial ideas that were successfully outlawed, without generating any significant violence (on the part of the government or the population) or civil unrest?
Apart from the obvious German anti holocaust denial / Nazi laws, there has of course been plenty of ideas banned without much of a reaction throughout history. One particularly widespread example is lèse-majesté laws, which criminalise disrespect towards the sovereign (or some other state authority). Such bans are found in numerous polities throughout history, ranging from ancient Rome to modern Thailand. Enforcement of such laws can sometimes have murderous results (e.g., Imperial China), but did not otherwise generate much violence or unrest.
(Of course, law enforcement generally depends on the use of force; so I am interpreting "significant violence" to mean a level of violence beyond typical policing.)
Perhaps the real difficulty with the question is defining "controversial". Lèse-majesté was widely accepted to be criminal as a matter of course for most of history. Likewise, modern bans on Nazism, or holocaust denial, or hate speech are only controversial because of libertarian concerns for free speech. These ideas are not in and of themselves controversial - the vast majority of people easily agree the holocaust happened, believe Nazis are evil, and abhors hate speech.
In fact, one challenge with this question is that if an idea gets banned without causing much reaction, then one could argue clearly it wasn't that controversial in the first place.
The few surviving instances of lèse-majesté laws still in force today could be considered an example. While they were widely accepted historically, for obvious reasons they are now commonly considered hopelessly antiquated. Nonetheless, it's still on the books of several countries, including even liberal democracies like the Netherlands or Denmark. However, their continuing prosecution of offenders does not appear to have resulted in any notable levels of violence.
Another example might be the suppression of the Hundred Schools of Thought in China. This was first attempted under the Qin dynasty, and then rather more successfully in the reign of Emperor Wu of the Han dynasty. Although the generally brutal Qin regime enforced the policy with characteristic violence, Emperor Wu was much more restrained - his ban primarily concerned only the public sector, and did not appear to have resorted to overt violence.
In both cases, no violent resistance occurred specifically in association with the policies. In Qin's case, this was likely because the people had much more pressing things to worry about, i.e. the brutal regime's heavy taxation - within a few years a series of peasant revolts brought the empire down. In Han's case, Emperor Wu made knowledge in Confucianism the key criteria for job advancement in his empire, and in this way let the opposing schools literally die out in obscurity.
Lastly, a possible example is the Great Schism of Christianity, which was rooted in disputes over some finer points of theology, such as the idea of papal primary. In 1054,
(representatives of) the Papacy and the Ecumenical Patriarch excommunicated each other over these and other differences (and politics).
Given the weight of excommunication in this highly religious era, one could say Western Christendom banned the Byzantine version of theology, while Eastern Christendom banned the Roman interpretation thereof. The underlying ideas are apparently controversial enough, that almost 1,000 years later, the two sides have yet to fully resolve their differences.
No significant violence resulted - in fact most contemporary Christians probably didn't notice the split at all. More importantly, support for each side of the controversy was split among geographic terms. That is, the finer points of theology in dispute was controversial in Christianity as a whole, but each half of Christendom largely supported the same interpretation, so the "bans", so to speak, didn't incite internal violence.
Catholic Crusaders did sack Constantinople in 1204, though this was not directly due to theology.
That said, all this is independent of the argument that:
outlawing an opinion won't get rid of it.
Which is technically true. That doesn't mean that the popularity of the opinion wouldn't be affected by a ban, though.
Consider again the ideological history of China. A vast array of ideologies once flourished during ancient China's warring states period, for example Mohism. In some ways it anticipated modern Communism, and at one point even rivalled Confucianism in popularity.
After Emperor Wu's reign, however, most of these schools of thought faded out of history due to the aforementioned government suppression. With the exception of Taoism which survived as a religion, and some of the more technical schools which persisted as technical manuals, China became wholly dominated by Confucianism up until the early modern period.
