I orginally thought that the U.S. had a quarter-cent coin when I was a kid, but I was mistaken. So, was there ever a country that had a quarter cent?
There's a few coins that's with the quarter denomination. Quarter kopek (polushka): Quarter ruble (polupoltinnik): Quarter yang (not really one cent)
The Straits Settlements (British colony) had 1/4 cent coins between 1845 and 1916. Ceylon also had 1/4 cent coins from 1870 to 1901 and French Indo-China in 1942/43. Probably some others out there too.