They did not. Not necessarily because the Greeks never noticed a connection, but because the rebirth of Greek writing and the Phillistines barely overlapped at all, and by the time they did there was very little connection left. The earliest Greek writing, like the Homeric poems and the works of Hesiod re-emerged in the 8th Century BCE after a long hiatus since the end of Mycenaean Linear B during the Bronze Age Collapse around 1200 BCE. The Philistines do not appear in any discernible form in those earliest mythological narratives, and around the time that Greek writing emerged they entered into a period of decline before "Philistine" disappeared entirely as a political or cultural identity around 600 BCE.
The primary literary source for the Philistines is the Old Testament of the Bible (the Jewish Tanakh), where they are primarily and enemy of the Hebrew kingdoms of Israel and Judah. Even in the Bible, the last direct reference to political events involving the Philistines is a passing remark about King Hezekiah defeating them in 2 Kings 18:8, sometime in the late 8th Century BCE, right around the same time Greek writing first reappeared. The surrounding verses of that passage provide some context for how the Philistines (as well as Judah and their neighbors) were in political turmoil at the time (7-12 in the link above so I can post in one part).
The rest of the chapter goes on to describe Hezekiah and Judah's conflict with the Assyrian king, Sennacherib, but this excerpt gives a glimpse into the political situation in the southern Levant at the end of the 8th Century BCE. The Philistines and their neighbors had been forced to pay tribute to the Assyrian Empire as it expanded into the region to secure a buffer against Egypt (then under the rule of the Kushites from Nubia). Prior to that time Egypt had either exacted tribute or pressured the Levantine cities to resist Assyrian influence. When Hezekiah resisted Assyria he also fought with the Philistines, who had submitted to the Assyrians.
The whole region remained largely under the Assyrian thumb for the rest of the Assyrian period, but when Assyria was conquered by Babylon and the Medes from 616-609 BCE the various peoples of the Levant began to reassert their independence. Near the end of that time frame, Pharaoh Necho II moved to expand Egyptian influence back into the region. He faced some resistance from independent minded Levantine rulers, like Josiah of Judah, but for the most part 609-604 BCE saw Egypt and Babylon compete for control over Syria and the Levant with Egypt even installing garrison in some Palestinian cities like Ashkelon. Ultimately though Babylon gained the upper hand and pushed Egypt back to its traditional borders, sacking Philistine cities (and Jerusalem) in the process. Ashkelon in particular is noted for having a distinct burn layer in 604 BCE, and then rebelling and being defeated again in 601 BCE.
The Bible has very little commentary on the Philistines in this period, as it is more concerned with similar events (backing Egypt initially and repeatedly failing to resist Babylon). After the return from Babylonian exile, the Philistines are no longer mentioned, but some of their cities are. For example, Nehemiah 13:23, references Ashdod as an independent group on the same level as the Ammonite and Moabites. Other cities remained important in the region. Ashkelon returned to prominence under Persian rule and Gaza was an important garrison. The running theme is that the region formerly identified by the Hebrews as entirely "Philistine" seemed to have become another cluster of independent city states by the 6th Century.
The Bible may also provide an indirect explanation for that. Nebuchadnezzar repeatedly quelled Philistine cities in the same period that Jerusalem was repeatedly besieged and defeated, leading to the first few waves of the Jewish exile in Babylonia. It's not unreasonable to extrapolate that the Philistines, especially the ruling class, may also have been deported. If there was not a successful "return movement" after the Persian conquest of Babylon, then that may be an explanation for the apparent lack of cultural continuity from 600 to 500.
All of that said, it is unclear just how much connection the Philistines maintained with Greece. It is an almost universally accepted conclusion that the the Philistines did come from Mycenaean Greece as part of the wider migration the Egyptians called "Sea Peoples." This is supported by the striking similarities in their material culture like pottery, a 2019 genetic study, and potentially some names and words in their language. Goliath, in particular, has been inconclusively connected with names like the Lydian Allyates or Greek Kalliades. The Philistines also traded with the Greeks during the early Greek Archaic Period. Their burials also mimicked Greek and norther Mediterranean funerary practices like pit graves and cremation, unlike the cave burials and ossuaries preferred in other Canaanite cultures. Excavations at Ashkelon have turned up large quantities of Greek pottery dated to the 8th-7th Centuries.
That said, these connections may have been very surface level, or potentially isolated to the ruling class. The aforementioned genetic study also showed that most of the bodies in Ashkelon did not show Greek or other Mediterranean genetics after the first few generations, seemingly indicating that the Philistines were largely absorbed into the local population. The same is true of the famous pottery connection, which left Mycenaean and Cypriot styles behind after 1000 BCE and took on more local Canaanite styles. All of the Greek pottery excavated at Philistine sites in the later Philistine period was imported from Greek cities in Anatolia or Europe.
The Philistine language is also poorly understood, mostly known from short inscriptions of names and Hebrew loan words that stand out as non-Semitic. However, the only inscription long enough to provide linguistic details is written in a local Canaanite dialect, similar to Phoenician.
The Bible also references gods worshiped there, including Baal and Astarte, but nothing that could be interpreted as a deity with Greek origins. The closest we get is Herodotus, 250 years after the end of the Philistine period, referencing a temple to Aphrodite in Ashkelon. However, he also references a temple to the same form of Aphrodite in Cyprus, which we know from Cypriot inscriptions probably means he was equating Astarte with Aphrodite.
That leaves surface level aesthetics like imported pottery and burial practices as the only confirmed, direct connections between Greek culture and Philistine culture for most of Philistine history. Those details alone probably would not have stood out as overwhelmingly similar.
That leaves just one curious legacy of the Philistines with the ancient Greeks: their name. How exactly the ancient Greeks drew this connection or learned the name is not documented, but by the 5th century BCE, they were calling the entire region from the Arabian desert to the Mediterranean and from southern Phoenicia to the Sinai Peninsula by the name "Palestine." Palestine was derived from the Egyptian and Semitic pronunciation of the word "Peleset," as the Philistines were known in contemporary sources like the Hebrew Bible or Egyptian inscirptions. Oddly enough, "Philistine" has the exact same etymology, just 400 years later. When the Tanakh was translated into the Greek Septuagint, Hebrew "Peleseth" was rendered as Greek "Philistine," reflecting the changes in how both languages were spoken over that time frame as well as a quirk of the translators.
Palestine remained the common name for the region in Greek and passed into Latin (when it wasn't being called Judea) and re-emerged when the Romans renamed and reorganized Judea as part of "Syria Palestina" in the 2nd Century BCE. Ultimately, "Palestine" stuck as a name for the region right up to modern times and the creation of Israel as a modern state, at which point it entered it's modern use as the state inside of/next to/underneath Israel.