Both Islam and Christianity have a long history of entanglement with what you might somewhat anachronistically call secular governments, and as a result both have a lot of violence in their histories. But the actual spread of the religions themselves, on a popular level, were generally non-violent.

Christianity initially spread through interpersonal interactions that were entirely non-violent, until the conversion of the Roman emperor Constantine, who legalized the religion in the early 4th century. After Constantine, Christianity's spread was aided by his official endorsement, but this spread remained non-violent until Theodosius made non-Christian religious observance a punishable offence at the end of the fourth century. There's not a lot of reliable evidence that pagans were persecuted following this change, however, and you really have to go to the sixth century with Justinian's attempts to force everyone in the Eastern Roman Empire to be his kind of Christian to see good evidence lf Christian-backed persecution (with steong political overtones).

Islam is a similar story. Muslim soldiers conquered the southern Mediterranean in the name of Allah, but they didn't force the people they conquered to convert. Instead, they replaced the local elites, levied taxes on non-believers, and alowed the Christians they ruled to keep their faith largely unhindered. Most historians agree it wasn't until the tenth century, more than 300 years after the first Muslim conquests, that most of the people under Muslim rule had converted to Islam through a gradual, largely non-violent process. Zoroastrians in Persia, who the Muslim conquerors regarded as pagans, received harsher treatment, but with that exception the spread of Islam, while politically violent, didn't involve violent forced conversions.

After that, the picture gets more complex, and you can find lots of religiously supported violence. But the i itial spread of both faiths was on a personal/community level, and not the result of violent coercion.