At the moment (today!!!!) probably not. be quiet does not have any suitable adapters for the 1200 socket at the moment. However, it is possible that in the foreseeable future (perhaps tomorrow) either the socket will be authorised or a corresponding installation kit will be added. But it could also be that a be quiet Dark Rock Pro 5 is in the pipeline. As a normal consumer, who knows what will be available next week.
The be quiet Dark Rock Pro 4 fan itself should certainly provide good cooling for normal use of an i9 10900k. I have one installed on my 9900k myself. Average temperature: 46°. Under extreme load (benchmark software) it reaches approx. 90°.

All 1151 coolers are compatible so this is not a problem, the dimensions are the same.

Ist kompatibel..... https://www.bequiet.com/en...