The specification of a "respectable" monastery made me giggle, but it's actually a great place to start. Sometime in the early/mid 12C, one of the most important and powerful monastic leaders of the era, Peter the Venerable (of Cluny), wrote a letter to one of the other most important and powerful monastic leaders of that era and many others, Bernard of Clairvaux. Peter had compiled a lengthy collection of Islamic and Arab-Christian texts, including the Qur'an, as the basis for "someone"--hint, hint--to write up a massive refutation of Islam. But, as Thomas Burman puts it, "Bernard did not take him up on the invitation."

So to provide a quick and dirty answer to your question: even Peter, the leader of one of Europe's most high-powered monasteries, knew his desired polemicist at another high-powered monastery could only understand the set of Arabic texts if they were translated into Latin--and that he had to hire an outside translator to do so.

But the story shows something a little deeper going on. Peter had Robert of Ketton translate the Qur'an in order to refute it--in other words, with a specific purpose in mind. Monastic learning wasn't ruled by, well, "ooh, shiny!" And the thing is--for almost every monk and nun behind cloister walls in the high and later Middle Ages, there was no need to learn Arabic. They'd never meet an Arabic-only speaker, ever, or travel to an Arabic-speaking land; the texts they prayed with and learned from were in Latin or eventually their local vernacular; they weren't going to be stepping into the shoes of medieval Europe's most idolized theologian, Augustine, and writing multiple treatises "against the heretics."

Hebrew knowledge was a little more common in the high Middle Ages, although certainly not to the "every respectable monastery" stage. This came about as Christians discovered that Hebrew commentaries on the Bible could provide insights for their own understanding. (In other words, they weren't learning Hebrew to read the Hebrew Bible/Old Testament text itself.) But understand that we're talking about extremely high-level academic work, not the equivalent of a weekly Bible study.

Especially with respect to Arabic, the location and reason of Arabic study shifted in the late 13th into the 14th centuries: from monasteries to universities, and from refutation alone to evangelization. The Church's 1312 Council of Viennes, best known for the condemnation of the Templars, laid it out:

In order, then, that skill in [lanuages spoken by non-Christians, namely Arabic and Hebrew] be attained by suitable instruction, we have stipulated, with the approval of the sacred council, that schools be established for the following languages wherever the Roman curia happens to reside and also at Paris, Oxford, Bologna and Salamanca: that is, we decree that in each of these places there should be catholic scholars with adequate knowledge of Hebrew, Arabic and Chaldaic.

There are to be two experts for each language in each place. They shall direct the schools, make faithful translations of books from these languages into Latin, and teach others those languages with all earnestness, passing on a skilful use of the language, so that after such instruction these others may, God inspiring, produce the harvest hoped for, propagating the saving faith among the heathen peoples.

Could this system actually be implemented at the universities or alongside them? Almost certainly not; the Council canon reflects an ideal, not the reality of the scarcity of teachers and interested students. But once again, we can see that the question of why learn Arabic was inseparable from learning it--and here, a belief or recognition of how few people did.

Nevertheless, we can see that the idea of languages in mysterious script were deeply compelling across the medieval world. Hildegard of Bingen, yet another of the most powerful monastic leaders of the era, is famous for a huge number of things, one of which was creating her own language (lingua ignota, or unknown language). In addition to it, possibly for it although the situation is murky, she also created--an alphabet.