Yes several can though perhaps not all. However, the most important one is HIV and this condition manifests as disease very late, not days or months after infection but years. HIV infection can be detected by the usual ERS (ELISA) tests after the window period of 12 to 24 weeks. Another condition, which may be detected in the absence of symptoms is syphilis, which can be detected by some blood tests. There are some other STI that can also be detected by serological blood tests.
In the absence of HIV infection you need to look out for conditions such as genital herpes, some form of hepatitis etc.
Personally, I do not think you will find it easy to completely rule out all Sexually Transmitted Infections (STI) with tests at this stage but you can of course rule out HIV with appropriate tests.