The short answer would be something like, "Well, if a song in a major key made you sad, then Yes."
A longer answer would have to trace the idea of "minor = sad, major = happy". If we are talking about Songs literally, then the question of the lyrics enters into it. But leaving that aside, we can go back to the ancient Greeks and find that they believed very firmly that different scales had different emotional effects. This notion can be found in the Baroque era as well. A scholar once told me "G Major was Bach's key of Spiritual Happiness"! And if you have any doubt about what Bach's key of spiritual despair was, listen to anything by him in e minor. Far more elaborate claims are made for the many scales used in North Indian classical music, which are associated not only with moods, but with seasons, time of day, etc.
None of which makes a conclusive case at all for these claims. Responses to music are so a) subjective and individual and b) strongly conditioned by culture, that there is no guaranteed response to any piece of music. So, the answer is: Maybe.