This is referenced in Cortés' Third Letter, late in the Siege of Tenochtitlan:

Since our power was running very short we had spoken over a fortnight before of making a catapult: and although we had no engineers really competent to undertake it, yet some carpenters offered to make a small one... The catapult took some four days to put into position even after it appeared in the square, Meanwhile our Indian allies were proclaiming to those in the city in what a marvelous manner we were going to kill them all... Both hopes however were doomed to disappointment, for neither did the carpenters succeed in working their engine, nor did the inhabitants though frightened make any move towards surrender, so that we were obliged to cover up the failure of the catapult by saying that moved by compassion we were unwilling to kill them all.^1

Cortés tends to be understated, but we have another account from Díaz del Castillo with some more color:

In Cortés; camp there was a soldier who said that he had been in Italy in the Company of the Great Captain and was in the skirmish of Garallano and in other great battles, and he talked much about engines of war and that he could make a catapult in Tlatelolco by which, if they only bombarded the houses and part of the city were Guatemoc had sought refuge, for two days, they would make them surrender peacefully... Cortés promptly set to work to make the catapult... When the catapult was made and set up in the way that the soldier ordered, and he said it was ready to be discharged, they placed a suitable stone in the sling which had been made and all this stone did was to rise no higher than the catapult and fall back upon where it had been set up... Cortés at once ordered the catapult to be taken to pieces.^2

Finally, we also have an account of this instance from the Mexica side of things, in Book 12 of Sahagún's history:

And then those Spaniards installed a catapult on top of an altar platform with which to hurl stones at the people. And when they had it ready and were about to shoot it off, they gathered around it, vigorously pointing their fingers, pointing at the people, pointing to where all the people were assembled at Amaxac, showing them to each other. The Spaniards spread out their arms, showing how they would shoot and hurl it at them, as if they were using a sling on them. Then they wound it up, then the arm of the catapult rose up. But the stone did not land on the people, but fell behind the marketplace at Xomolco.

Because of that the Spaniards there argued among themselves. They looked as if they were jabbering their fingers in one another's faces, chattering a great deal.^3

All in all, this is a relatively well documented event during the war between the Mexica and the Spanish-Tlaxcalans. It fits with the Spanish having to rely on items like gunpowder being delivered from the coast and having to improvise. Some projects, like the brigantines constructed by Martín with Tlaxcalan labor and the utilization of native craftsmen to make crossbow bolts were fairly successful, but the catapult is recognized as an unequivocal failure, with only who to blame for its construction and failure being in question.

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^(1 Cortés H [trans. JB Morris 1969] Five Letters of Cortés to the Emperor, p. 218)

^(2 Díaz del Castillo B [trans. D Carrasco 2008] The History of the Conquest of New Spain, p.298)

^(3 Sahagùn B [trans. J Lockhart 1993] We People Here: Nahuatl Accounts of the Conquest of Mexico, p. 230)