I agree with the above post. There are several conflicts present in The Tempest because there are a few subplots within the course of the play. All have to do with some type of sociopolitical hierarchy that the characters fall into during their lives. As mentioned above, the main conflict revolves around Prospero attempting to get justice for having been usurped by his brother, but there are other conflicts as well such Stephano and Trinculo attempting to have a type of power that they would never be allowed anywhere else. Similarly, Caliban is oppressed by Prospero who finds Caliban to be a base creature. At the end of the play, many conflicts are resolved, yet it is unclear with Caliban's fate will be. Ariel has been set free, but as far as the reader/viewer knows, Caliban is still Prospero's slave.
Of course, this question is purely subjective, and the answer depends entirely on what you see in the play. I disagree that the play contains no conflict. It perhaps may not follow the "norms" for what happens in Elizabethan/Jacobean drama, and many people think that the play was Shakespeare's farewell to the stage. Still, the play is character-driven, mainly by Prospero, and it holds plenty of conflict.
Prospero experiences conflict with nearly every other character in the play, from his daughter Miranda, who would love to get off the island, to Caliban, his unwilling servant, to Ariel, his willing servant, who yet wants to be freed from servitude.
The story began with a long-ago conflict that caused Prospero to be banished with his daughter. He is, at the time the play takes place, trying to bring a resolution to that old conflict, not just by settling a score, but by providing for Miranda's future. True, things may work out fairly smoothly, but that does not mean that conflict does not exist in the play. It only means that Shakespeare was a master at dramatic resolution.