How much is a quarter of florin cents from 1859 worth?

You've got to be more specific.

First, the usual definition of a florin would be the British coin worth 2 shillings (1/10th of a pound). Because of this, there would be no such thing as a "quarter of a florin" let alone "cents" which cents have never been used when referring to British currency.

So a quarter of a florin would be half a shilling or sixpence. But sixpence coins would always say sixpence and never would say "a quarter of a florin"

I'd look at your coins again and post a new question.