tl;dr

This is not a real story but an illustrative description, probably invented in the 1930s.

The first Indo-Arabian numerals came to Europe in the 10th century. They had a hard time at first. In the 13th century the Italian Leonardo Fibonacci published the Liber abaci (1202) which popularised their use further, but mainly in Italy. In 1522 Adam Ries then published Rechenung auff der linihen und federn in Germany – and in German instead of Latin.

Hypothesis:
That is the pretty well established spread of that numerical writing and calculating system in Europe. That anecdote in question is intended as an illustration for that. It is doubtful that this exact conversation took place or was recorded in that way. After all, why did "the eminent professor" not teach that child himself, if he was familiar with Indo-Arabic numerals and convinced of their advantages?

After all, using Roman numerals is cumbersome compared with Indo-Arabic numerals, for us. But multiplications can be made with an abacus and that instrument is suited for the use of Roman numerals.

Then we have to look at the university system at the time. The artes liberales did include arithmetic and geometry in the Middle Ages. But that was fine even with Roman numerals. And neither Medici, nor Fugger, nor Welser merchants went there for study to learn the business.

They learnt by doing business (or in their own, rare, lay, abbacus schools.) One most prominent example, Jakob Fugger the Rich:

A document from the Austrian state archive has now shown that Jakob Fugger was already representing his family business in Venice in 1473 at the age of 14. Other research showed that Jakob Fugger spent the years between 1473 and 1487 mostly at the Fondaco dei Tedeschi, the house of German merchants in Venice. Venice being one of the most important centers of trade at the time proved to be an ideal environment for Jakob Fugger's education in banking and the metal trade. His long residence in Italy also helped bring the renaissance style to the German region, with his funding the construction of the first buildings of this style that originated in Italy. Legal and architectural structures of Venice also had a significant influence on the funding of the Fuggerei which was similar to the social housing of Venice.

So the term "university" seems to be the give-away. Without it, the anecdote looks plausible. Some merchants did sent their sons to Italy, they sent them there for learning the trade, as the Italian system was way more advanced in the high Middle Ages than anywhere else in Europe (banking). The universities were also old and good in Italy. Just that merchants didn't go there.

And why should they? Mathematics education was quite the step-child in European universities, being very impractical in nature:

We may perhaps wonder why the medieval university, with all its success in the domains of logic and natural philosophy, and in spite of the activity of several noteworthy mathematicians, never brought it far in the domain of mathematics education. […] As far as mathematics is concerned, lectures combined with discussion favour the development of metamathematics – that is, also philosophy. But in order to become creative in mathematics itself, and possibly to enjoy it, one has to do mathematics, not only to speak about it. Inside the curriculum of the learned schools and the universities, the areas where one could do mathematics were few. Computus was one such area – but its mathematics did not go beyond simple arithmetical computation. Rithmomachia was another one, and the game indeed remained popular until the sixteenth century. The third was computation with Hindu-Arabic numerals in the use of astronomical tables – perhaps not too inspiring either, but nonetheless a domain that was practised assiduously well into the Renaissance, whether for its own sake or (rather) because it was a sine qua non for simple astrological prediction. […]
We know very little about the education of burghers’ children after the twelfth-century revival of city life. A few institutions like the Saint Victor school in Paris admitted them, but what they offered seems to have been badly adapted to a future in commercial life (future artisans were in any case taught as apprentices); Pirenne (1929, p. 20) relates that a Flemish merchant’s son was put into a monastic school around 1200 in order to learn what was needed in trade – but then became a monk. Some clerks served as house teachers in wealthy families (Pirenne 1929, 21ff), and some probably held private schools.
That Italian merchants had been taught by Latin-writing clerics is illustrated by Boncompagno da Signa’s description (1215) of their letters as written in a mixture of corrupt Latin and vernacular.21 Computation was presumably learned on the job, during apprenticeship – but even this is nothing but an educated guess built on what we know from later times.
_{Jens Høyrup: "Mathematics Education in the European Middle Ages", in: Alexander Karp & Gert Schubring (Eds): "Handbook on the History of Mathematics Education", Springer: New York, Heidelberg, 2014.}

The same anecdote is re-told (p14) in Frank J. Swetz & David Eugene Smith: "Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith", Open Court Publishing, 1987. They weren't able to authenticate the anecdote either.

But the origin is traceable at least to Tobias Dantzig: "Number. The Language of Science", MacMillan, 1930 (archive.org, p27). There we find no source attribution either, but an important qualification:

There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young man was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained.
As a matter of fact, multiplication and division as practiced in those days had little in common with the modern operations bearing the same names. Multiplication, for instance, was a suc- cession of duplations, which was the name given to the doubling of a number. In the same way division was reduced to mediation, i.e., “halving” a number. A clearer insight into the status of reckoning in the Middle Ages can be obtained from an example. Using modern notations:
We begin to understand why humanity so obstinately clung to such devices as the abacus or even the tally. Computations which a child can now perform required then the services of a specialist, and what is now only a matter of a few minutes meant in the twelfth century days of elaborate work.
The greatly increased facility with which the average man today manipulates number has been often taken as proof of the growth of the human intellect. The truth of the matter is that the difficulties then experienced were inherent in the numeration in use, a numeration not susceptible to simple, clearcut rules. The discovery of the modern positional numeration did away with these obstacles and made arithmetic accessible even to the dullest mind.

That would make the anecdote a tale of morality, with bits and pieces found in history cobbled together to form an instructive story of slow but triumphant progressivism,
which is unfounded in actual history, as evidenced by too many wrong details in the story (and all its unsourced variations (example, set even earlier, but getting it very wrong in the process).

Like another instructor concluded:

The state of science in medieval Europe can be characterized through an anecdote reported in Ifrah (2000):

A German merchant wanted to give his son the best possible education. He called for a respected professor and asked him to which university he should send him. The professor's advice was: "A German university will do if he only wants to learn addition and subtraction. If he wants to learn multiplication and division as well he should go to an Italian university."

Anecdotes are like caricatures; they exaggerate typical features, but they have a true core. The story of the medieval merchant demonstrates that spending an entire lecture on medieval science in Europe is an undeniable act of cultural bias. From the point of view of global history it cannot be justified. The only excuse I can offer is that I was born into the European civilization and therefore have an interest in even the darkest times of European history.

_{Matthias Tomczak: "The state of science in medieval Europe.", Science, Civilization and Society, CPES 2220: A course of 35 lectures, first given at the Flinders University of South Australia during the second half of 2004.}

Curiously, this anecdote speaks of a German merchant. The English speaking web and books from the 1930s onward recount this tale numerous times, mostly with just miute variations.
Yet German books seem to copycat this story only in very recent years. Not that it would count for anything, but the earliest record for this in German language publications seems to be from 1999 (and that one even being an American originally)?