I've been treated with methotrexate 2 sets of 2 shots and was never read or made aware of such a side effect. Not sure that is one... have you checked WebMD or typed up rare methotrexate side effects?
Methotrexate Side Effects | Drugs.com
...from what I've seen *photosensitivity* is a much more common side effect rather than tanning easier. Which can occur with allot of medications, chemo drug therapies and so much more. (like anti-biotics) So I'm not sure what to tell you, but I agree call your practitioner and tell them your concerns. When in doubt get it checked out!