Jesus in this passage, Matt. 10:34-36, is not suggesting that family conflict is normal or desirable. To hold this position would conflict with his command to love one another, and the fourth commandment. However, Jesus is predicting that his message and teaching will cause conflict when one person in a household comes to faith and others do not. This would be a serious problem in both Jewish and Gentile communities in the early church. See e.g.. St. Augusta.
It can also be a problem today in unchurched families, or within families whose members are only superficially Christian. In this case, as one person begins to live the full implications of the faith it can very easily make others uncomfortable.
In the middle ages, if a child wished to become a professed religious, and be celibate, this could cause conflict within the landed nobility because of arranged marriages and the inheritance of the lands and titles. See the story of St. Almedha.