No. Nor does it originate with other venerable authors commonly implicated, Plotinus, Aquinas, Ficino, etc. Plato's "triad", as read into Philebus, was supposedly Truth, Beauty and Proportion, Good was the highest form uniting all three. The OP linked site presents some other "creative" pseudo-ascriptions of it to Plato involving Republic and Symposium. But the current pop-culture coinage comes from Cousin's Lectures on the True, the Good, and the Beautiful (1853). Cousin had precursors, such as
Konig's treatise on good taste (1727), translated by Rollin into English in 1770, where we read:
"The general good taste is an intellectual capacity, derived from sound witsand a keen power of judgment, to correctly experience the true, good and beautiful"
Another one was Diderot, who referred to it repeatedly from 1747 on, e.g. in the 1782 Essay on the Reigns of Claudius and Nero he writes;
"Where would we be, if perverse men could turn false that which is true, bad that which is good,or ugly that which is beautiful? The true, the good and the beautiful form,in my eyes, a group of three great figures, around which evil can raise up a storm of dust that may conceal them from the eyes of the majority, but the time passes, the cloud dissipates, and they reappear as venerable as ever."
Its further spread in the 18th century proceeded through writings on aesthetics by Riedel, Herder, Tetens, Kant and Schelling. However, Cousin, a neo-Kantian, divorced the triad from forms of taste as their common bond in man's nature, and connected it instead to the disjoint Kantian faculties of reason, will and sensibility, unified only by postulating God as the source of all three.
Martin traces the developments in The Birth of the True, the Good, and the Beautiful:
"In 1853 Victor Cousin published his Lectures on the True, the Good, and the Beautiful, a compelling statement of his eclectic philosophy, one that he believed could be summarized as a pursuit of these three values... A number of wonderful ironies lie here in his adoption of what I will henceforward call “the” triad of the True, the Good, and the Beautiful... Cousin used it to preach a sentimental theism, when this triad was formed only through the rejection of theism... I will briefly trace the key processes that led to the development and consolidation of this triad, which has had a remarkable holding power over Western thought since the mid-eighteenth century.
"It is often assumed that there is some precedent for the triad in the works of the ancient Greek philosophers, which is quite untrue. There was, however, an intellectual nugget that was to play a major role in the development of the triad, namely the idea that some persons might have a certain type of beauty/nobility and goodness... It is also often assumed that the triad harks back to the medieval doctrine of the transcendentals, but this is equally untrue. The “transcendentals” were terms that were coextensive with Being thus to say that Good is a transcendental is to say that Being, insofar as it is being, is good (e.g., Aquinas,Truth Qu 21, art. 1, 1954, pp. 3-6)... “Goodness” and“Truth” only appear as different because Being enters into different relationships with the human soul and mind. Although Beauty might appear along with Truth, Goodness, and a whole host of positive terms (as in the works of Pseudo-Dionysius, 1987), the triad of transcendentals was always that of the One, the True, and the Good...
At the end of the medieval period, the great distinction was that made between the intellect and the will - what Ficino (see Platonic Theology Book II.1.1, 1.14 V.8.8, IX.1.3, XIV.2.2, XIV.3.5.6;2001, pp. 93, 97;2002, p. 85;2003, p. 11;2004, pp. 227, 229, 247) called the “two Platonic wings” on which the soul ascends to the Divine - though this bifurcation was often mapped onto two different triads. One was the transcendentals (leaving “the One” without a human correlate), and the other was the more orthodox triad of the attributes of God as being Wise,Good, and Powerful... Ficino is often misremembered as originating the triad of the True, the Good, and the Beautiful, either in his commentary on Plato’s Symposium, or in his commentary on the Philebus; neither of these is correct. The closest he comes is in the latter...
The irony of the story of the triad, then, is the following: from the work of neo-Kantians like Cousin, we now have a sense of the True, the Good, and the Beautiful being different dimensions, indicating inherent incommensurable directions for the pursuit of value. We often also believe these three to be mutually exclusive and exhaustive of value in itself. Yet we have seen that the triad could only stabilize when this was denied - when truth was beauty, and beauty was goodness... If it was only God’s nature that held the three together as fundamentally “one and the same,” the rejection of this “hypothesis” would lead to a splintering into radically disjoint dimensions that lacked any consubstantiality. And this is exactly what happened in the generation after Cousin’s... Indeed, when the triad survived... it was only by being transmogrified into “values,” a new notion of quasi-transcendentals that continually failed to transcend, and continually frustrated attempts at a unified anthropology".