That may have already happened in real life.
(Emphasis on the "may", because this is not a 100% mainstream accepted theory, but it's good enough for fictional inspiration.)
There are many languages that have special registers for specific social or ceremonial circumstances, in which large sections of vocabulary, grammar, or even phonology may be replaced relative to the common register--e.g., the "Ja" registers (or "mother-in-law" registers) of several aboriginal Australian languages. These are typically used for taboo avoidance, but in traditional Dyirbal society there are specific arguably-religious ceremonies which demand the exclusive use of Ja by at least some parties, and it has been seriously suggested that the ceremonial register may in fact be an intentional conlang.
Why would they, or your monks, do something like this? Simple: to draw a another boundary between the sacred and the profane. Religions, especially the sort that produce monks, are permeated cross-culturally by a common distinction between the worldly or profane and the spiritual or sacred, and religious practice often involves a symbolic crossing from the profane world into the sacred world, either by literally crossing into a sacred space (like a church, temple, or monastery) or by the adoption of specialized religious clothing, physical rituals (e.g., ritualized washing)... or ritualized speech patterns.
If your monks are at all familiar with the idea of constructed languages, or can come up with it themselves (as historically at least one actual nun has done--St. Hildegard of Bingen, whose Lingua Ignota unfortunately did not catch on in the larger monastic community but was in fact explicitly designed as a sacred language for religious devotion), it would not be at all implausible for them to decide to adopt a new language along with all other aspects of their new religious life as just one more component to set them, their sect, and their worship apart from the rest of the world, and give them a unique sense of community.