I was once told that Sparta had the world's first professional military. Is this accurate?

No, for at least two reasons.

Sparta did not have a professional, standing military, it had a leisure class that was obligated by law to engage in some amount of training. Our resident Classical Greek warfare sage /u/Iphikrates details the nature of the army in this classic series of posts.
The Spartan military apparatus of the Classical era was long preceded by the reforms of Tiglath-Pileser III, an 8th-century Great King of Assyria and perhaps the most important military reformer between that time and Philip II of Macedon. Tiglath-Pileser reformed the traditional system of raising a seasonal army from the home province of Assyria, incorporating subjugated peoples directly into the army (a practice that would be used to great effect by later Assyrian monarchs as well as the Achaemenids), allowing for effective year-round campaigning and the drafting of soldiers around a permanent (or at least more long-term) core. He's known to have introduced a series of administrative reforms, and must have reformed the taxation system as well, but the precise nature of these reforms is not well understood.

As in every other case, whether Tiglath-Pileser's reforms resulted in a truly "professional" or "standing" army is highly contentious; as with the Achaemenid armies, the bulk of the troops were no droubt drafted as needed. T.L. Davenport's 2016 PhD thesis on the subject notes:

Assyrian sources attest to the existence of two main military bodies during our period: the kiṣir šarrūti or ―royal cohort which was stationed in the capital and fell under the direct command of the Assyrian king (Postgate 2007, pp. 348-349), and the ṣab šarri or ―king‘s men which operated in the provincial territories under the authority of the governors (Postgate 1974, pp. 219-223). The precise role played by each unit in the operation of empire, as well as their individual status, is problematic and difficult to determine on the basis of the available evidence (see Fales 2010, pp. 140 ff.). While it may be tempting to view the kiṣir šarrūti as comprising the core forces of the army and the ṣab šarri as simply reinforcements (see Fuchs 2011, p. 387), the evidence, particularly from the reign of Tiglathpileser III onwards, does not support this conclusion and reveals that the organisation and operation of Assyria‘s forces was far more complicated.

[Postgate] argues that this term is misleading, since it suggests that only the kiṣir šarrūti were battle-ready year round, when in fact the ṣab šarri was also, technically speaking, a ―standing army‖, since it too operated on a permanent basis year-round (2007, p. 351). Yet, most troops were conscripted under the ilku-system from both Assyria and the provincial territories on a temporary basis and thus were not strictly speaking a ―standing army, since they only operated seasonally (Postgate 1974, pp. 223-224). However, it is clear from the evidence that some soldiers, both Assyrian and foreign, served in the army in a more permanent capacity (2007, p. 346 & n. 29, p. 347).

The evidence seems to point toward the existence of a core group of permanent soldiers in both Assyrian armies, as well as an administrative apparatus for raising additional troops as needed. Hence, it can not be considered professional quite in the sense of the Imperial Roman or Macedonian armies, but here one must also account for the existence of fundamentally different needs, socio-economic constraints, and military tactics - the Assyrian army being heavily missile-based.

In conclusion, if we regard army professionalization as existing on some sort of sliding scale, it's hard to see the Spartans as trailblazers in this respect.