This also famously appears in Tolkien and works derivative of his.

'Every worm has his weak spot'

This is indeed the quote from Tolkien's Hobbit (Chap. 12), and it is also true that several lore of the dragons across Eurasia allot their weak spot as stated in this alleged proverb.

Problem is, however, on which spot of the body part European dragons often have/ or lack their scale defense, and whether the movie adaptation really follows this tradition.

The critical moment goes in Seamus Heaney's verse translation of Beowulf as following:

'Next thing, they say, the noble son of Weohstan

saw the king in danger at his side flames and strikes

and displayed his inborn bravery and strength.

He left the head alone, but his fighting hand

was burned when he came to his kinsman's aid.

He lunged at the enemy lower down

so that his decorated sword sank into its belly

and the flames grew weaker.

Once again the king

gathered his strength, and drew a stabbing knife

he carried on his belt, sharpened for battle.

he stuck it deep into the dragon's flank.

Beowulf dealt it a deadly wound' (ll. 2697-2705).

The weak spot seemed to be located on the dragon's belly (either hit by Wiglaf and/or by Beowulf).

Fáfnir in Old Norse tradition, slain by Sigurðr, also seemed to share this weak spot that lead to his heart, as stated as 'Þá lagði Sigruðr hann með sverði til hiarta (Then, Sigurðr pierced the dragon with the word to its heart)' in the preface of Fáfnismál (Neckel & Kuhn hrsg. 1983: 180).

In later Volsunga saga, an old man (Óðinn) advised Sigurðr to pierce Fáfnir's heart, and Sigurðr follow this advice by striking on the dragon's left side

• 'An old man with a beard came up to him and asked what he was doing told him. "That's ill-advised," the old man then replied. "Dig other pits and let the blood run into are to sit in one and stab the dragon to the heart.'

• 'And when the dragon crawled across the pit, Sigurd thrust under the left shoulder, and it sunk in up to the hilt......' (Volsunga saga, Chap. 18, in: Finch ed. & trans. 1965: 29, 31)

I suppose that Tolkien built Smaug and his weak spot mainly based on these traditions, but I'm not specialized in Tolkien and his literature, I'm not so confident with Smaug itself.

On the other hand, however, a weak spot 'under the throat' might have originally belonged to totally alien tradition on European dragons. An Ancient Chinese text commonly attributed Han Feizi (3rd century BCE), Chapter. 12, compares the dragon with the lord, and advises not to dare to touch such a 'inverted scale' of the lord as well as the dragon when one wish to persuade the lord:

'Indeed, when the dragon moves like a worm, man can tame it, play with it, and ride on its back. However, there are below its throat inverted scales, each one foot in diameter. These scales would kill anyone touching them. So does the lord of men have inverted scales. Any persuader able to avoid touching the inverted scales of the lord of men must be very near to the mastery of the art of persuasion'. (The translation of Han Feizi is taken from this site)

So, if OP's memory as well as description of the body part of the weak spot is really accurate, the dragon in question might have come from Eastern Asia.......

References:

• Kuhn, Hans & Gustav Neckel (hrsg.). Edda: Die Lieder des Codex Regius nebst verwandten Denkmälern. 5. verbesserte Aufl. Heidelberg: Carl Winter, 1983.
• Finch, R. G. (ed.& trans.). The Saga of the Volsungs. London: Nelson, 1965.