There are diamonds in meteorites, but they are very small, and not of gem quality. Diamonds in meteorites are produced in two different ways, from two different sets of material.
Collisions between asteroids can be so violent and generate such high shock pressures that carbon is converted to diamond. The meteorite diamonds produced in this way are very cracked, and only one to two millimetres in size. These are solar system diamonds, as they were produced directly within asteroids.
The second type of diamond in meteorites is found in rocks that have not been shocked, or even heated. These diamonds are tiny: only about three nanometres across (three millionths of a millimetre).
It is thought that they might have formed when a nearby star exploded as a supernova. So they are from before the solar system was born.