Investigating this question I found the following on where these supposed fragments of Papias are located:
George Hamartoulus wrote in the 9th century: "John the Apostle after he had written his Gospel suffered martyrdom, for Papias in the second book of the Logia Kyriaka says that he was put to death by Jews, thus plainly fulfilling along with his brother the prophecy of Christ regarding them, and their own confession and common agreement concerning him"
An epitome of a fifth century work by Philip of Side says: "Papias says in his second book that John the Divine and James his brother were slain by Jews" - this is however someone quoting him, saying he said Papias said this.
Both are quite late sources, and Clement said somewhere that not all the Apostles suffered martyrdom. These aren't definite quotes in my opinion, and the original works of Papias from which it is supposedly culled is lost.
I think the only strong evidence for John being martyred is Mark 10:39, but I find this ambigious, and thus not strong at all.