The claim is wrong on all counts. The help requested from Athens was not disaster relief; the Athenians did not refuse; while the ensuing events were a key moment in the changing relationship between Athens and Sparta, it would take another 33 years before the Peloponnesian War broke out.

The episode is not very well attested, but the basic facts seem to be these. In 465/4 BC, Sparta suffered a disastrous earthquake. Diodoros claims that more than 20,000 people died, while Plutarch says all but five of the houses in the town were destroyed. When they heard of the catastrophe, the helots and some of the perioikoi rebelled in an attempt to destroy Sparta for good. But the Spartans under king Archidamos managed to organise themselves for their defence just in time, and eventually the insurgents were driven to a stronghold on Mount Ithome in Messenia. The Spartans, unable to take the stronghold, called on their allies for help. At the time, the old alliance against the Persians still held, and Athens was an ally of Sparta. The Athenians therefore sent an army to Messenia to help the Spartans crush the helot revolt.

However, on the arrival of the Athenian army, the Spartans got nervous and sent them home again, alone of all the allied contingents. The Athenians took this rejection of their sincerely offered help as a serious insult. They renounced their alliance and their pro-Spartan policy, ostracised the pro-Spartan commander Kimon, and forged new alliances with Argos and Thessaly.

It is not clear exactly why the Spartans sent away the Athenian force. Thucydides (1.102.3) and Diodoros (11.64.2) claim it was because they feared that the democratic Athenians would be tempted to side with the rebels and betray Sparta. Plutarch (Life of Kimon 16.8) adds that the Athenian reformer Ephialtes actually opposed the idea of sending help, but Kimon convinced the Assembly to support their ally; the Spartans still rejected their aid, but only because they feared Athenian boldness (17.2). Neither explanation seems very persuasive, but any others are just speculation. Perhaps Sparta didn't want to be seen relying on Athens to take care of its problems, especially now that Athens had clearly become the ascendant power in the Greek world.

Either way, even if Thucydides claims this was "the first open quarrel between Lakedaimonians and Athenians" and Diodoros says "the Athenians took this incident as the first cause of the estrangement of the two states," the Spartan rejection of Athenian aid was neither the first crack in the alliance nor the immediate cause of their later war.

As early as 479/8 BC, when the Persians had only just been defeated, a conflict arose between Athens and Sparta because Athens began to rebuild its city wall. The Spartans much preferred Athens to remain defenceless, so they protested against the works and tried to prevent them; the Athenians sent Themistokles to play for time and keep the Spartans distracted while they got their walls built up enough to allow them to defy Spartan demands. Later on, when Thasos revolted from the new Athenian Empire in 465 BC, Thasos sent to Sparta for help against Athens, and Sparta was willing to oblige. In fact it was the earthquake that stopped them from declaring war on Athens (Thuc. 1.101.2).

But the disaster was serious enough to turn the Spartans away from further adventures. The alliances Athens made in the late 460s were troubling, and war did break out in 461/0 BC (the so-called First Peloponnesian War), but this was primarily a conflict between Athens and Corinth. Sparta focused its efforts on keeping its local rival Argos in check. The armies of Athens and Sparta clashed only once, at the battle of Tanagra (457 BC), which only happened because the Athenians intercepted a Spartan force on other business. The defeat Sparta inflicted on Athens at Tanagra was not decisive. It eventually became clear that neither side could be easily made to yield, and a thirty-year truce was signed in 446/5 BC.

The eventual all-out conflict known as the Peloponnesian War (431-404 BC) arose out of a number of other causes. While it's possible to see earlier traces of animosity as far back as 479 BC, and Sparta's rejection of Athenian aid after the earthquake was a major turning point, the immediate conflicts of interest that led to the outbreak of the Peloponnesian War all date to the later 430s BC. It is not possible to connect the Spartan decision to go to war directly to the events at Ithome more than three decades earlier, and no source tries to make such a connection.