Is there any authentic hadith, specifying how prophet zakariya (peace be upon him) die ?
You ask for an authentic hadith on a topic that did not take place in the time of the prophet (p.b.u.h). Doing this, you assume that this information has been revealed to the prophet (p.b.u.h). But even if Muhammad (p.b.u.h) mentions it, he may have had the information from prior tradition or scripture. Together with the imprecisions that come from hearsay, I think that hadith is not likely to be more reliable than the original related information.
It is related in 2 Chronicles 24:20–22 that Zakharia was killed.
Then the Spirit of God took possession of Zechari'ah the son of Jehoi'ada the priest; and he stood above the people, and said to them, "Thus says God, `Why do you transgress the commandments of the LORD, so that you cannot prosper? Because you have forsaken the LORD, he has forsaken you.'" But they conspired against him, and by command of the king they stoned him with stones in the court of the house of the LORD. Thus Jo'ash the king did not remember the kindness which Jehoi'ada, Zechari'ah's father, had shown him, but killed his son. And when he was dying, he said, "May the LORD see and avenge!"
The book of Chronicles is contained in the Tanakh, among the canonised books of the Jews. Canonised means that it has been considered reliable to some extent and it is protected from any later changements. Of course, there was no system established to evaluate reliablilty of the contents as we have it for the hadith.
Zakariah father of John is an other person who lived centuries later. The tradition of the murder does not refer to him and sources stating that are probably mixing up the two homologous persons.