No.

Poison and disease certainly played a role in Greek warfare. At the battle of Plataia in 479 BC, the Persians made the initial Greek position untenable by poisoning the Gragaphian Spring, the Greek army's only water supply. The city of Syracuse was repeatedly saved from capture because their enemies (whether Greek or Carthaginian) were forced to encamp near marshy ground to besiege the city, and were inevitably gripped by epidemics. During the Peloponnesian War (431-404 BC), the Athenians initially decided to withdraw within their walls and abandon the countryside to the enemy, but in the packed conditions of the city, a devastating plague of unknown nature broke out, severely reducing Athens' manpower reserves. When the Spartans heard about this plague, they wisely decided to restrict their regular invasions of Athenian territory until it blew over. There is also the famous story of the mercenaries of the Ten Thousand, who, during their retreat through the Persian Empire in 401/400 BC, found and ate some 'mad honey' and wandered around hallucinating for a day. Modern scholars have argued that the honey in question was probably made from yellow-flowering rhodondendron luteum, containing acetyl-andromedol, which causes the symptoms described by Xenophon.

However, none of these examples involve the deliberate application of biological weapons with decisive effect. Often the involvement of factors such as disease was accidental; cases where poison was used on purpose were not decisive.

If you'd like to read more, there's a book on this very topic: Adrienne Mayor's Greek Fire, Poison Arrows, and Scorpion Bombs: Biological and Chemical Warfare in the Ancient World (2003).