Yes, because it's an ability explicitly linked to ki.
Nothing in D&D 5th edition can be read in isolation, everything has a context. By narrowly trying to interpret a section of text, without considering that context, we can follow an incorrect path of logic.
The Monk class is all about mastery of ki (emphasis mine):
Whatever their discipline, monks are united in their ability to magically harness the energy that flows in their bodies. Whether channeled as a striking display of combat prowess or a subtler focus of defensive ability and speed, this energy infuses all that a monk does.
Monks make careful study of a magical energy that most monastic traditions call ki. This energy is an element of the magic that suffuses the multiverse—specifically, the element that flows through living bodies. Monks harness this power within themselves to create magical effects and exceed their bodies’ physical capabilities, and some of their special attacks can hinder the flow of ki in their opponents. Using this energy, monks channel uncanny speed and strength into their unarmed strikes. As they gain experience, their martial training and their mastery of ki gives them more power over their bodies and the bodies of their foes.
As a result of the structured life of a monastic community and the discipline required to harness ki, monks are almost always lawful in alignment.
The Tongue of the Sun and Moon feature states (emphasis mine):
Starting at 13th level, you learn to touch the ki of other minds so that you understand all spoken languages. Moreover, any creature that can understand a language can understand what you say.
The ability to both understand all spoken languages, and having any creature understand what you speak is specifically linked to your learned ability to touch the ki of other minds.
In fact, from the text of the ability, it's clear that this isn't some latent ability of your body that you have awakened, but an ability triggered by you actively touching the ki of other minds that you have learned to control.
[Monks] as they gain experience, their martial training and their mastery of ki gives them more power over their bodies and the bodies of their foes.
As a monk, your whole class and features are defined by your mastery over ki, as a result of careful study [of ki], and your incredibly disciplined approach to that study, and your life. If you cannot control whether or not you are touching the ki of another mind to facilitate them understanding you, then you can't really be described as a master of ki no can you?
As a result, we must conclude that you, as a monk, have the ability to choose whether or not this ability is active.