A Monk won't be able to "Automatically" resist the charm effects of any Sentient Items that try to control them. The Stillness of Mind feature doesn't specify any kind of resistance or immunity to charm effects, it only provides an at-will means of undoing the effect.
So in practice, what does that mean?
The Monk can end a Charm Effect—Any Charm Effect—whenever they want, so long as they want to
That last part is important, for a number of reasons.
The Monk might not know they have been charmed
This depends on a lot of factors, including how much Metagaming is permitted at your table (i.e. can a character make a decision based on what the player knows, but which the character couldn't possibly know?), and how much information the DM conceals from players. The Monk can't choose to end a Charm effect on themselves if they are unaware that they have been Charmed.
With Sentient items, there's a pretty strong 'tell' when a Sentient Item tries to take control of its user, so the Monk may very well be aware that they have been controlled by the item. In that case, it would take very little effort to remove the Charm effect. Unless...
Effects like Dominate Person—or the control exerted by a Sentient Item—can forbid the Monk from removing the effect
The 5th level spell Dominate Person—which inflicts the Charmed condition on the target—has the capability to take complete control of a person:
You can use your action to take total and precise control of the target. Until the end of your next turn, the creature takes only the actions you choose, and doesn't do anything that you don't allow it to do. During this time you can also cause the creature to use a reaction, but this requires you to use your own reaction as well.
It also stipulates on the things you can order it to do (without taking complete control), which includes
While the target is charmed, you have a telepathic link with it as long as the two of you are on the same plane of existence. You can use this telepathic link to issue commands to the creature while you are conscious (no action required), which it does its best to obey.
In the former case, it's quite obvious how Dominate Person can prevent the Monk from using Stillness of Mind. In the latter case, it would require some adjudication from the DM, but an order like "Do not try to remove the effect of my spell" might be sufficient to prevent the Monk from ending the effect.
For Sentient Items, the Monk will run into a similar problem as the latter situation, as stipulated by the description of how a Sentient Item might try to take control of its wearer:
If a sentient item attempts to take control of its wielder, the wielder must make a Charisma saving throw, with a DC equal to 12 + the item's Charisma modifier. On a failed save, the wielder is charmed by the item for 1d12 hours. While charmed, the wielder must try to follow the item's commands. If the wielder takes damage, it can repeat the saving throw, ending the effect on a success. Whether the attempt to control its user succeeds or fails, the item can't use this power again until the next dawn.
It's not difficult to see how the same issue that they might face with a Dominate Person caster would pop up here as well. The Sentient Item would only need to stipulate "don't try to remove this Charm effect!", and the Monk would be compelled to obey that order.
Sentient Items are... Sentient
In both a positive and negative sense.
A Sentient Item probably doesn't know all the secrets of the universe. It might know that its wearer can attempt to resist the item's control, but it might not know just how well the wearer can do so. Even if they control a Monk, and the Monk ends the effect immediately afterwards, the item might not connect the dots and work out "Oh, I need to order them not to remove the effect; I'll try again tomorrow!". It might just conclude "well rats, I guess I can't control them!"
So on one hand, the Item might not be clever enough to find a loophole to exploit; on the other hand, it might fail to properly control its host one day, and then another day find the right phrasing or buttons to push that will let it get what it wants. It's dynamic, which means its wearer will never be able to fully predict the item's behavior.
In Conclusion, it's not safe for a Monk to wear a Sentient Item
It is definitely safe-er than it would be for other party members, simply because against most items, they'll have at least one free "get out of possession free" card they can deploy if the item turns on them—which would be their cue to take the item off and never use it again.
But it'll never be safe to the point that the Monk can disregard the danger wholesale—at least not as a general principle to be applied to all Sentient Items. Some Sentient Items might have more reified behaviors and quirks, which may make the item safer—or more dangerous—than the average. But as a general rule, a Monk should not behave like they're simply immune to an Item's attempts to manipulate them.