It depends on what the caster is doing.
I think the answer to this is in what dominate monster actually does. Nothing in dominate says the target views its former allies as enemies; only that it obeys instructions.
Let's look at the specific effects of dominate monster:
The target is charmed by the caster. A charmed creature can't attack the charmer, and the charmer gets advantage on social rolls, but that's all. It doesn't have any impact on the target's opinion of his allies.
The caster can issue commands that the target will attempt to obey, such as "attack that creature". But obeying the instruction doesn't mean they consider their allies to be enemies. The dominate effect is forcing them to do the task as instructed, but they're not necessarily going to do anything hostile other than what they were generally commanded to do.
And finally, the caster can use an action to take total control of the target. Let's look closer at that one.
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.
So here is the first place where I see anything that would stop the target from willingly failing a save; the target "doesn't do anything that you don't allow it to do", so that would potentially include choosing to fail a save, where that's an option, or being a willing target of a spell.
So it looks to me like the victim of a dominate spell could intentionally fail the save on calm emotions, but only as long as the caster isn't assuming direct control.