For diamonds to be available to us on the moon, we need several things to have happened:
1. Availability of carbon
We don't have much information about the abundance of carbon in the moon. A cursory examination of the amount of carbon available in Earth's crust and the Apollo samples indicates that carbon is ten times more abundant in Earth's crust than the moon's. However, Apollo barely scratched the surface of the moon; all we know about what's beneath the surface comes from a few rocks that were blasted up by meteor impact,and then picked up by the Apollo astronauts.
2. Formation of diamonds
On Earth, diamonds formed under conditions of temperature, pressure, and local gravity gradient which existed one to three billion years ago, perhaps as deep as 120 miles (200 km) beneath the surface. To put that into perspective, that time scale puts us in the Middle to Late Proterozoic period of Earth's development, when life was just barely getting organized.
Although 120 miles sounds deep, it is still close enough to surface of the Earth that the local gravity would be 1 g for all practical purposes. That works both ways: the high gravity conditions tend to inhibit formation of large crystals, but also means that 120 miles of rock on top of the diamond-forming region raise the pressure to levels that squish carbon molecules into a regular crystalline structure.
The time scale could be very important. Earth cooled very slowly in comparison to the moon. Diamonds that crystallized one billion years ago still had about three billion years to get organized. During that period the great smelter of the molten planet had time to sort out all sorts of elements, forming a densitometer on planetary scale. One layer of the densitometer is plain carbon atoms. In theory, there may be a thin spherical surface laden with diamonds just 100 miles beneath our feet.
Although Earth had a very long time scale for forming diamonds, we know that billions of years are not required. We produce artificial diamonds in laboratories in hours rather than eons. So the time factor and rate of cooling observed on Earth does not preclude diamond formation on faster-cooling planets such as the moon.
There's always the very likely possibility that, as with terrestrial geology, pockets of carbon (perhaps from carbonaceous chondritic asteroids) were trapped beneath layers of rock during the moon's formation. This might have created conditions of pressure and temperature necessary to form diamonds.
Based on experiments with creating artificial diamonds, the moon would need temperatures of at last 2,732 deg F at a pressure of 975,000 pounds per square inch to create a diamond. Did any of those carbon molecules get squished at that temperature and pressure in the geological history of the moon? We really can't know until we get there and start digging.
3. Transportation to the surface
The moon might be full of enormous diamond crystals, but they won't do us much good if they're not close enough for the surface for us to get to them.
We find diamonds near the surface of Earth mostly because of volcanic activity. Magma from below punches trough the carbon layer where the diamonds formed and carries them to the surface. The volcanic tubes left behind when the magma cooled are called kimberlite pipes, after the Kimberly Mines in South Africa. Plate tectonics also play a role in transporting deep material to Earth's surface.
The moon's history appears to be a combination of violent volcanic events and lava flows from massive impacts. We can only speculate whether these events were sufficient to move enough material close enough to the surface to provide the lunar community with diamonds.
The only solid conclusion we can draw is that we don't know enough about the moon to draw a conclusion.
Suggestions for Further Research
For more in-depth research, see the list of publications of the American Geophysical Union at http://www.agu.org.
You'll find some great on-line resources from the Department of Geology and Geophysics at the University of Wisconsin, http://www.geology.wisc.edu/.