It's unlikely. And even if there is, they've not announced it publicly. Most of the private companies and startups in this area are still in the stealth mode. This is the most complete list of quantum computing startups that I know of. Among the companies listed, Atom Computing may be working on diamond-based quantum computers, but they haven't released much information yet.
All I could find about them is: "Utilizing techniques pioneered by quantum gas and precision measurement experiments, Atom Computing will create machines using optically trapped neutral atoms capable of reaching the large qubit numbers needed for error-corrected quantum computing." For what it's worth, one of their co-founders Jonathan King specializes in this area and was a co-author on Multi-spin-assisted optical pumping of bulk 13C nuclear spin polarization in diamond (Pagliero et al., 2018).
You may also be interested in the article Toward mass-producible quantum computers published by MIT news, which discusses the current progress in NV-center based quantum computing. Now why diamond-based quantum computing isn't so mainstream is a good question, but you should probably ask that in a new thread.