Reacting only to the title you selected for this thread, before I read the question itself: There are few if any medical therapies that affect the reliability of HIV testing. The only drugs that might do so are very potent immunosuppressive drugs in high doses, or potent chemotherapy of the sort used for widespread cancer. Even those are theoretical only, with few if any actually documented cases of interference with HIV diagnosis.
Now I have read the question itself. Laquinimod is a new, experimental drug that might have promise for multiple sclerosis and perhaps other immunological disorders of the nervous system. I very much doubt it would interfere with standard HIV antibody testing, including the oral fluids test, since almost nothing does. However, to my knowledge there is no research on this. In situations involving new, experimental therapies, the investigators doing the research are by far the most knowledgeable about the drug. You need to ask the researchers responsible for your research study. I will be surprised if they are concerned about interference with your OraQuick test, but please let me know if they say otherwise.
You don't describe the nature of your possible exposure to HIV. But barring a surprise from your research team, you can be sure you didn't catch it. Whether or not you should have still further testing depends on the risk: the lower the chance of catching HIV, the less important it is to test out to 6 weeks or 3 months. Use the search link to find innumerable other threads on the topic of seroconversion window, i.e. time to positive HIV tests.
Best wishes-- HHH, MD
**Correction**
Doctors -- Please forgive the additional post below but I want you to have all the information I do. Turns out I was also being taking Moxifloxacin as well -- somewhere around the same amount for the same duration. Please consider this when giving me advice. I appreciate it.