(covers information from several alternate timelines)
A character reference was a recommendation of an individual by those who knew them. It attested to their abilities and virtues.
In 2258 of the alternate reality, Spock came onto the bridge of the USS Enterprise and informed James T. Kirk of his intention to be considered for the position of first officer. He offered to provide character references, but Kirk didn't see the need. (Star Trek)
When Benjamin Sisko referred to Quark as the community leader for Deep Space 9 in 2369, Odo drily dispelled the Ferengi's disbelief by saying he had all the character references of a politician. (DS9: "Emissary")