How can I make a USB cable as narrow as possible?
You haven't given any details about the air quality monitor. Judging from the picture, it's a simply DC power supply. So the data wires most likely aren't used.
Therefore you can go ahead as proposed:

Cut the wire
Thread it through the hole
Reconnect the 5V and ground wires

There is no need to reconnect the data wires or the shield (which shields the data wires).
You can also use a different wire with two conductors, e.g. to extend it. It should be sufficiently thick to carry the current, in particular if it is extended.