The only source I've been able to find for this claim is the Historia Augusta, a collection of biographies of emperors and people connected to them. In the biography of Caracalla from this collection, we get the following anecdote:

It is not out of place to include a certain mockery that was uttered at his expense. For when he assumed the names Germanicus, Parthicus, Arabicus, and Alamannicus (for he conquered the Alamanni too), Helvius Pertinax, the son of Pertinax, said to him in jest, so it is related, "Add to the others, please, that of Geticus Maximus also"; for he had slain his brother Geta, and Getae is a name for the Goths, whom he conquered, while on his way to the East, in a series of skirmishes. (12.5)

In other words, Roman leaders were given names or titles based on peoples or places they had conquered; Caracalla had killed his brother Geta and had also conquered the Getae, and Pertinax Jr. joked that Caracalla should be called "Geticus" because of it. You will notice this passage does not say that Pertinax Jr. was killed for this. In fact, a different part of the biography of Caracalla states that Caracalla killed Pertinax "for no other reason than because he was the son of an emperor, and he would never hesitate, whenever an opportunity presented itself, to put to death those who had been his brother’s friends" (4.8).

But this is not the only place in the Historia Augusta where this comes up. In the biography of Antoninus Geta, we read approximately the same story of the pun, immediately followed by the sentence: "This remark sank deep into the heart of Bassianus, as was afterwards proved by his murder of Pertinax, and not of Pertinax alone, but, as we have said before, of many others as well, far and wide and with utter injustice" (6.6). (Bassianus is another name for Caracalla.) This passage is still not quite saying that he was killed for the pun: the text both situates the killing in the context of a larger crackdown against potential political enemies and states that Pertinax Jr. was not killed until later, but it does present the pun as something that put Pertinax Jr. on Caracalla's bad side.

So, does this mean that it really happened? The exact origin of the Historia Augusta is debated; it's not even clear whether the various biographies were written by one person or several. It seems to have been written around 400 CE (though this is debated), so we're looking at a source approximately 200 years after Caracalla's reign. It is also notoriously untrustworthy as a historical source, though given how many texts are included in the HA, it varies from one to the next. There are anecdotes that are obviously fictional and read like novels. There's a list of imperial pretenders of whom many never really tried to usurp the throne, and at least one probably never existed. The life of Caracalla is one of the less bad ones, but by default, you should not assume anything in the HA is true.