It is possible to find diamonds in the ocean. Diamonds are typically formed deep under continents and can be brought closer to the surface by a type of eruption known as a kimberlite. Over time, kimberlite deposits can be eroded, but since diamonds are so hard, they resist weathering. These diamonds are washed into rivers that eventually carry them to the ocean. De Beers, one of the world's biggest diamond producers, has a joint operation with the government of Namibia called DeBmarine Namibia. Debmarine ships scrape or vacuum up lots of sediments from the seafloor where robotic submarines have detected diamonds. The diamonds are then filtered out of the sediment and collected. This method tends to find high quality diamonds, since flawed diamonds tend to break apart before reaching the seafloor.

Unfortunately, it is also quite destructive to the underwater environment because animals can get caught, and the clouds of sediment being released can be bad for life that needs clear water. De Beers and Namibia say that the seafloor environment can be observed to recover itself, but the long term effects are still not known.

Diamonds can be found in the ocean, but they do not form there.

Most natural diamonds found on Earth are formed in Earth's mantle layer, around 200 miles below the surface. At these depths, pressures and temperatures are high enough to cause the carbon in sources such as CO₂ to bond in the specific atomic arrangement known as diamond. The diamonds are transported to the surface when the liquid(ish) rock of the mantle is thrust upward during volcanic eruptions. At the surface, the molten material cools and solidifies, trapping the diamonds inside a material called Kimberlite. Over time, weathering breaks down the Kimberlite and exposes the diamonds. Finally, rainwater carries them to rivers which gradually wash them out to the oceans (which is an example of erosion, subtly different from weathering). (As an aside, volcanoes do occur in the oceans, but I was unable to find whether these were a significant source of oceanic diamonds. There are also a couple other ways by which diamonds are formed on Earth, but these contribute a relatively small fraction of the total.)

In recent years, technology has become sufficiently developed for companies to take advantage of this source of diamonds. Around 2002, the De Beers company (which historically has dominated the diamond industry) and the government of Namibia began a joint venture to scavenge diamonds from the seafloor off the coast of Namibia. They use autonomous and sometimes manned submarines to locate regions containing diamonds, dig and suck up sediment from the seabed, then sift out the diamonds. One of the concerns with this approach is the damage to the local marine environment; this type of offshore mining disturbs or removes parts of the seabed which are vital to many species and can take years or decades to recover.

Diamonds usually form deep inside the earth. This is because you need to put Carbon in very high temperature and very high pressure to form it into a diamond. Big volcanic eruptions bring diamonds to the surface of the earth. These high pressures and temperatures can take place deep under the ocean floor, and diamonds could be released into the ocean during eruption of underwater volcanoes. In fact, rare "blue diamonds" that contain some Boron in addition to Carbon are thought to form under the ocean floor- here is an article about how blue diamonds form: formation of blue diamonds.

If we look at the theories of how diamonds form, we see that the proposed places where diamonds form are not very close to ocean floors. That is not to say that the diamonds that formed can't somehow end up in the ocean, but that's not very likely. To answer your question: It's possible, but not very likely.

Diamonds are found in rocks. There is no reason why the type of rocks that contain diamonds cannot be underwater, but diamonds do not form in the ocean.

1. Cool - diamonds are formed from what used to be the ocean floor long long ago. But most of those diamonds are now found in mines on land.

Most diamonds are made of cooked seabed. The diamond on your finger is most likely made of recycled seabed cooked deep in the Earth. Traces of salt trapped in many diamonds show the stones are formed from ancient seabeds that became buried deep beneath the Earth's crust, according to new research here.

2. Scientists think a lot of good diamonds blasted out of volcanoes long ago, and they were washed into rivers and the to the ocean. So there's a new diamond industry, searching for diamonds at the bottom of the ocean. But people are worried that mining for diamonds in the bottom of the ocean is terrible for the ocean and the life that lives in it.

"A new frontier for diamond mining: The ocean. OFF THE COAST OF NAMIBIA — Deep beneath this frigid stretch of the Atlantic Ocean, some of the world's most valuable diamonds are scattered like lost change. ... But those mines are gradually being exhausted." Source.