I know 10 tevet is a minor fast day. I saw reference to a opinion that 5th tevet should be observed as a fast?
n 422 BCE, the prophet Ezekiel heard the news of Jerusalem’s conquest by a foreign power, as recorded in the biblical Book of Ezekiel (33:21). In Jewish law, a period of mourning (for example, upon the death of a loved one) can begin upon “hearing” the bad news. Some Talmudic commentators thus recommended that the 5th of Tevet be instituted as a public fast day.
Is this observed today?
It isn’t observed. There is an old sefer called Megillas Taanis, wriiiten even before the Mishna was wriitten, and it has numerous days that tragic and happy things happened to the Jewish people. Nowadays we don’t observe any of these days, rather only the days that mentioned in the Shulchan Aruch.