This works in theory
The conjure fey spell does not impose the charmed condition, the creature simply obeys you. Thus it can be affected by charm monster and gain the charmed condition from it. When your concentration breaks, the charm will still be in effect, preventing the creature from attacking you.
There are a few things that have to be noted:
- The charmed condition does not explicitly protect your allies, although it does give you Advantage to persuade the creature not to attack them.
- The creature has to fail the saving throw of charm monster.
- This will depend on your DM, but you might not be able to tell whether they failed the save or not. The creature is already obedient, being charmed might not be noticeable.