Two Miis usually start on the ordinary friendship track. The rough flow is: strangers, slightly familiar, seem to get along, close, quite close, very close, and unbelievably close. Once the relationship reaches a certain point, you get a chance to change what they call each other. If you miss it, do not worry; you can still change names later in the Mii speech settings.
If you want A and B to get closer, you need to create more chances for them to interact. Let them chat, live near each other, or have A help B when B is frozen, fallen over, choking, or asking for help. This is not a 100% button, but it is the main way relationships rise naturally. Some players even go to extremes and put them on a tiny isolated island so they have more chances to run into each other.
One thing to watch: in the game, parent-child affection and romance are different routes. If you set A and B as parent and child in the real-world relationship settings, then let them live together, the game may confirm that family link through the news. After that they can move through parent-child affection stages instead of romance stages. In that case, the two Miis will not develop a romantic relationship, which also makes it a way to avoid a pairing you do not want.