Can the HIV show up in normal routine check-up blood work?

No. You need to ask specifically for the HIV blood test. Some blood tests may alert the doctor that something is wrong. HIV-infected individuals may have abnormal blood tests, and there are certain abnormalities that are commonly seen in infected people. Should these abnormalities be found, it would be similar to the doctor finding clinical signs in a physical examination that would raise his suspicion. In areas where the prevalence of HIV is low, such as the USA and Europe, it is still unlikely that the cause is HIV - there are many causes for these abnormalities. Even in areas where the prevalence of HIV is high, these abnormal results cannot be used to decide whether or not someone has HIV - they still are often caused by something else, just as swollen lymph nodes are often caused by something else.