We need verse 7 to verify that verse 6 refers to end-of-time events. Job uses the imagery of storm-driven winds as part of the final winnowing:
Job 21:17-18
17 How oft is the candle of the wicked put out! and how oft cometh their destruction upon them! God distributeth sorrows in his anger.
18 They are as stubble before the wind, and as chaff that the storm carrieth away.
The imagery Ammon uses is more vivid, however. It is possible that Ammon’s imagery of storms and fierce winds relies upon known conditions in that area of the world rather than the Old Testament text. Mesoamerica is an area that has been struck by many destructive hurricanes. It is quite possible that this imagery of destructive and powerful winds might be a particularly poignant mode of final destruction in an area of the world that has seen the devastating power that can come with a hurricane.
Regardless of the imagery, what Ammon is saying is that the Anti-Nephi-Lehies have been truly converted, and that their conversion will withstand all of the storms of the adversary. Their faith has already withstood the threat to their physical lives, and Ammon is sure that it can withstand any assault on their spiritual lives.