Conversion was a change in one's core beliefs, brought upon by examination of the evidence, or in some cases, coercion against one's will. One who experienced such a change was called a convert.
Joseph, the SS Santa Maria's engineer, was the last convert to the technology-free lifestyle practiced by the colonists of the planet Orellius. In 2370, while trying to convince Benjamin Sisko to remain on Orellius and adopt their lifestyle, Alixus withheld water and sleep from him, saying she didn't expect a "sudden, miraculous conversion" out of him, but she did hope for a show of good faith such as changing out of his Starfleet uniform. (DS9: "Paradise")
In 2375, Kasidy Yates-Sisko declined to perform a ceremony to bless the Bajoran women looking to be mothers on Deep Space 9, something Sisko, as the Emissary of the Prophets, had done up until his marriage to Yates. She explained to him that she hadn't converted to Bajoran religion when she married him and didn't want to fake it. He understood, saying he wouldn't want her to do anything that made her uncomfortable. (DS9: "Strange Bedfellows")