No, probably not unbalanced
The rule on Improvised Weapons states in part:
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage [...]
Depending on what object your monk friend found to use as an improvised weapon, it is entirely possible that it could be treated as a monk weapon. In this case, it would be 100% balanced because the rules explicitly state this is permissible.
If the found weapon is not a close proximity to a monk weapon, the answer is probably still yes-adjacent. Improvised weapons use a d4, which is a very low damage die, and identical to a dagger (which is a monk weapon).
As Someone_Evil points out, monks get to change the base damage of monk weapons:
- You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die [increases] as you gain monk levels.
As a result, all improvised weapons will increase in their base damage to match the monk's unarmed strike damage so, in the end, the monk is not gaining any real advantage in using the improvised weapon compared to making unarmed strikes.