Blood testing is done for some musculoskeletal conditions, but it's usually to rule out an infectious or inflammatory condition. There are no definitive blood tests of the type you are looking for.
A few studies have been done looking at blood results during the acute phase of whiplash. They found slightly raised levels of blood markers three days after a whiplash injury. These changes were back to normal within two weeks' time.
Blood markers are substances found in the blood that can indicate a particular disease process. For example, tumor markers are used in oncology to help determine the presence of cancer. An elevated level of a tumor marker can indicate cancer; however there can often also be other causes of the change. For some types of cancer, it's possible that a tumor marker could be identified for someone with cancer that had metastasized to the bones of the neck.
Other tests of blood markers found no differences in people with neck pain exposed to work-related repetitive motion. More studies are needed to find out if a blood test can predict or identify various conditions associated with neck pain.