Only if they are actually testing specifically for cancer. They wouldn't pick it up if they were just testing for something common (toxicology for eg).
There are so many cancers.........but the simple answer is YES.
A routine blood test would pick up the many types of Leukaemias.
A PSA for men is now becoming a routine part of a blood test for men to exclude prostate cancer.
Tumour markers are very specialised and have to be requested only at specialised units.