That story seems like nonsense, and I suspect your teacher is confusing the incident in Augustine's Confessions (VI.3) where he describes Bishop Ambrose as always reading silently. Augustine seems impressed, but I've always taken it to be Ambrose's intensity and focus which impressed him. He mentions that visitors would come to see Ambrose when he was reading like this and that Ambrose, rapt in study, wouldn't notice them. It is, as far as I'm aware, the first mention of someone reading silently exclusively, though there are some scattered mentions of silent reading in other classical texts. Thus, it appears like the practice wasn't especially common, but it wasn't a magic trick. Augustine's reaction isn't "Holy Mother of God, how can he even do that?!?" but "Wow, Ambrose is very studious and learned!"

Edit: Here's the relevant Augustine passage. The "he" is Ambrose:

Now, as he read, his eyes glanced over the pages and his heart searched out the sense, but his voice and tongue were silent. Often when we came to his room--for no one was forbidden to enter, nor was it his custom that the arrival of visitors should be announced to him--we would see him thus reading to himself. After we had sat for a long time in silence--for who would dare interrupt one so intent?--we would then depart, realizing that he was unwilling to be distracted in the little time he could gain for the recruiting of his mind, free from the clamor of other men’s business. Perhaps he was fearful lest, if the author he was studying should express himself vaguely, some doubtful and attentive hearer would ask him to expound it or discuss some of the more abstruse questions, so that he could not get over as much material as he wished, if his time was occupied with others. And even a truer reason for his reading to himself might have been the care for preserving his voice, which was very easily weakened. Whatever his motive was in so doing, it was doubtless, in such a man, a good one.

No -- categorically and unequivocally no. Up until 1997 it was generally accepted that reading was normally aloud in Greco-Roman antiquity, but that view was comprehensively dismantled in a pair of articles that came out that year. It is no longer tenable in any way or to any extent. It is simply false.

The old consensus was based almost solely on a single passage in Augustine's Confessions 6.3, which is presumably the one you're thinking of in your reference to the fourth century. In that passage, Augustine describes intruding on Ambrose's reading and sitting in silence, and pondering Ambrose's motives for sitting in silence so long and wondering why he was reading to himself. The old view is not only a misreading of the Augustine passage, it also neglects literally dozens of older pieces of evidence on the subject.

As I said, the counter-argument wasn't put together until 1997, so if you found this view in a source written before that point, the author shouldn't be censured for being wrong -- just out-of-date. Here's part of an old thread that goes into details. As far back as we can look, there are many, many more references to reading that imply silent reading than those that imply reading aloud. Some of them are as explicit as the Augustine passage. The thing that has kept Augustine at the top of the citation list for so long is that the Greeks in particular routinely used metaphors of "speaking" and "hearing" for what texts say.

Here are the earliest relevant references, all from the fifth century BCE:

• Euripides, Hippolytos 856ff. - Theseus opens a letter to "see" what it "speaks" to him; a few lines later (and there's no evidence that he leaves the stage) he cries out at its horrible contents, stating -- using the aural metaphor -- "it shouts, it shouts". At 882 he cries, "I will no longer keep this unspeakable story within my mouth's gate".
• Euripides, Iphigeneia at Aulis 34ff. and 107-108 - Agamemnon reads and edits a letter repeatedly, but a slave standing next to him still doesn't know its contents.
• Herodotos 1.123-125 - Cyrus is alone, so the references to the letter "saying" things and Cyrus "hearing" them are the aural metaphor again; cf. an inscription "speaking" in Hdt. 8.22.
• Aristophanes, Knights 115ff. - Nikias gives Demosthenes a bundle of oracles to read, and Demosthenes is so absorbed in reading them that when Nikias asks him what they say, he says "pour me another cup of wine". A few lines later he paraphrases what's in the text without quoting it directly.

There are many, many more in the 1997 articles: if you want to track them down you can find the references in the thread I linked above.

Not an expert by any means, but I've come across a resource in my studies that might be useful: Paul Saenger's The Space Between Words: the Origin of Silent Reading (1997)

In the book he formulates an argument that the introduction of blank spaces in writing fundamentally changed Western comprehension of text. In the classical world, text was written in a style called scriptura continua, meaning, essentially, that manuscripts were large blocks of text with no word separation.

Word division, Saenger says, originated in Insular manuscripts in the 8th century and was largely a result of translating Latin into Celtic and Germanic vernaculars. "It wasn't literally impossible to read silently," says Saenger, but the notation system made it illogical, or perhaps just more difficult, to do so. He suggests that instead of thinking that people "discovered" silent reading, it is perhaps more useful to view the way that we read as "modern reading," which makes more sense with our current notation system.

To answer your question a little more, perhaps the specific monk in question is Bishop Ambrose. In Saint Augustine's Confessions (4th century), he speaks about how Bishop Ambrose would read with "his voice and tongue at rest," although Augustine says that he was more revered for his devotion to the reading than the feat of silent reading itself. (http://www.bostonleadershipbuilders.com/austin/book06.htm#3)

Another source (just to definitively rule out that the people of antiquity couldn't read silently) is Plutarch's Lives of the Noble Grecians and Romans, which includes an anecdote about Julius Caesar reading silently a note that was slipped to him during a conversation with Cato the Younger. Although we can't tell if the anecdote is factual, Plutarch's Lives was written in the late 1st century, so we can safely say that the idea of reading silently was at least known in Plutarch's time.