Circuit's Load Falling Below Voltage Regulator's Quiescent Current
what will happen to the voltage regulator if the MCU goes to sleep. Will it cease to produce any current on its output?

It will provide the 5uA demanded by the MCU at the specified voltage.
Quiescent current will flow from the input voltage source to the &quot;GND&quot; pin of the regulator.

my voltage regulator has a quiescent current of 50uA

Also note that this current varies (a lot) according to the load and temperature:

The IC has a shutdown pin, but it seems to not be an option for your application since the &quot;sleep mode&quot; of the MCU is being used (instead of a reset).