So first off your picture of the supposed conflict between Christianity and science in the West is problematic. I'm not really going to get into that here (except to say the conflict myth is largely just that, a myth). In any case it's been covered many times here and is something that's fairly easy to look up yourself, and ultimately that isn't the question you asked. Now just because I've started off by downplaying the popular notion of conflict between science and religion, I don't mean to encourage you to adopt something like Gould's view of the two subjects as "nonoverlapping magisteria," that's equally problematic.

As for your question. Yes there have been incidents of "conflict" between religions other than Christianity and science. Considering that the only other answer is promoting the widely held belief that Buddhism is and has been largely compatible with science, that's a good place to start.

As Tibet started to see more western missionaries during the 20s and 30s, debates were staged where Buddhist monks could be seen defending the earthly existence of Mt. Meru and its accompanying flat Earth cosmology which had been a part of Tibetan Buddhism for quite some time (yes many, but not all, Tibetan Buddhists believed the Earth was flat well into the 1920s). Tibetan monks moved to adopt western science as something like a defense mechanism against the attacks of missionaries for whom science had become an effective tool of colonization. Theosophy also promoted this picture of compatibility as it dragged Buddhism out of its cultural context, westernizing it, mystifying it, and generally building in fundamental complementarity with science before exporting "Buddhism" back to the East as a philosophy. Even the current Dalai Lama, despite expressing great appreciation for science, has effectively rejected aspects of modern evolutionary theory so as to preserve the relationship between rebirth and karma. With the notable exception of the current Dalai Lama and his opinions on evolution, it's really a story of the violence science has done to Buddhism.

EDIT (comment was deleted so this point is without context) I also take issue with the idea that there is anything like "the scientific method" as would most historians of science, but that can be argued over another time.

If you want to read about the historical relationship between Buddhism and Science, check out Lopez-"Science and Buddhism" It's well researched but still a pretty poor book. The literature here is pretty sparse. For Islam Saliba-Islamic Science and the Making of the European Renaissance gives a very nice account of the relationship between Islamic culture and science during the Islamic Golden Age (it may be a bit technical for your purposes).