The original poster's problem was not with the word "quarter", but with
the word "of". The phrase "a quarter of nine" is completely
meaningless... what exactly is one-fourth of nine o'clock, anyway?
9:15? But "a quarter of nine" actually means 8:45.

It's not meaningless, and it's not that unusual. "Of" is used that way in a
lot of expressions -- "He's within 100 miles of Calgary." "That guy is two
bricks shy of a load." "Of" denotes proximity. A quarter is understood to
be fifteen minutes, for obvious reasons. A "quarter of" an hour is "fifteen
minutes proximity" of an hour. It's an old expression, I know, but it's not
all that uncommon around here (Wisconsin).