Can you put an 18 gear back wheel on a 21 gear bike?

It is possible to put a 6 speed back wheel on a 21 gear bicycle but you will never be able to use more than the 6 gear allowed by that cassette sprocket.

The best way to take advantage of the 21 gear bicycle is to change that 6 speed cassette sprocket and replace it by a 7 speed cassette sprocket. This way you will keep that back wheel and also be able to use all 21 gears of your bike.