This rumor started after a report claimed inaccurately that the spike-shaped protein on this coronavirus was the same as another protein called syncytin-1 that is involved in the growth and attachment of the placenta during pregnancy. It was quickly debunked as false by the scientific community, but had already been widely circulated on social media. The messenger RNA in these vaccines never makes it into the center of your cells, where your DNA is kept, and there is no evidence at all that the vaccine affects fertility.

SOURCE: https://www.bu.edu/articles/2021/myths-vs-facts-covid-19-vaccine/ (Boston University)

Updated 12/16/2021