Scotland has gradually changed to using English without being conquered by England. Now I shall temper that a little in that there have been English speaking people living in Scotland just as long as there have in England. The Anglo-Saxons settled south east Scotland as well as north east England. However, English didn't become the majority language for hundreds of years after that.
Scotland had several native languages such as Scots Gaelic, Pictish, a Brythonic language similar to Welsh, and also some Norse or Norn speakers. The Angles brought in Old English. Magnus Magnusson's book "Scotland, The Story of a Nation" has a map of this distribution on page 35.
Old English changed to Middle English and then split to become Scots and Modern English. By the late middle ages just three languages remained. Gaelic, Scots and a small minority English speakers. (Map of Languages in Scotland circa 1400)
After the Union of Crowns and latter Union of Parliaments, Modern English started to increase in use and Scots and Gaelic withered. However, the Union wasn't an invasion. Scotland was never conquered by the English. In fact the opposite. The Scottish king inherited the English crown.
Today almost nobody uses Scots. Instead they use Scottish English with a few hits of Scots. There is a small group of Gaelic speakers but no where near the number of just 200 years ago. Wikipedia gives numbers of 297,823 speakers (18.5% of total population) in 1800 and just 58,652 (1.2%) today. Thus I would argue that Scotland has changed it's language without being conquered.
Mongolia would be a contender for a second answer. Mongolia split from China during the early 20th Century. At that this they used a mix of Mongolian and Chinese (Mandarin) and a little Tibetan. They had there own Mongolian script for writing. However, today if you travel to Mongolia, you will see little use of the Mongolian script. It was replaced with Russian Cyrillic script. The strong influence of of the Soviet Union, which bordered Mongolia, brought about a change in the written language of the area. However, although the Soviet Union had a strong influence, Mongolia always remained a separate nation state and was never conquered by the USSR. Today Mongolian script is usually only seen in Inner Mongolia, within China. Outer Mongolia changed.
Taiwan would also make a third answer. This one maybe politically contentious depending on how you view the status of Taiwan. However, before 1949, Mandarin Chinese was a minority language on Taiwan. When the Republic of China government relocated to Taiwan in 1949, a large number of refugees from the Mainland settled there. Today Mandarin Chinese is the majority language. So officially, the language changed without invasion.
Some Taiwanese will not agree with my assessment as they consider Taiwan to have always been independent form China and was assigned to Chinese control, and occupied after WWII. However that is a minority opinion. Others will disagree with me on the opposite argument that Taiwan is not an independent country and thus as part of China, doesn't count.
Singapore will be my fourth contender as an answer. The official languages are Chinese, English, Malay and Tamil. The Last two, Malay and Tamil, are the native languages of the are. English came in via conquest and colonial rule. However, Chinese is now the largest language in use (49.9% of total population) but Singapore has never been conquered by the Chinese. Rather a large influx of Chinese immigrants has changed the cultural balance of the country.