Standard American English - no. "my finger pains" isn't correct. (However, there are some dialects of English that do say that. And everyone would understand you. Just most people in the US would think that it sounds wrong.)

A quick Google Ngram search of British and American English for "my leg/back hurts" and "my leg/back pains me" shows that "my leg/back pains me" is specific to American English and relatively rare. https://books.google.com/ngrams/graph?content=my+leg+hurts%2Cmy+leg+pains+me%2Cmy+back+hurts%2Cmy+back+pains+me&year_start=1800&year_end=2000&corpus=17&smoothing=3&share=&direct_url=t1%3B%2Cmy%20leg%20hurts%3B%2Cc0%3B.t1%3B%2Cmy%20leg%20pains%20me%3B%2Cc0%3B.t1%3B%2Cmy%20back%20hurts%3B%2Cc0

"My finger pains me" is rare and a bit archaic or dialectical, but I've seen it. I've never seen "my finger pains" without a direct object. "My finger hurts" is much, much more common.