Yes, that's true.
(Assuming you have a standard 20-button in C/G, which your question suggests is the case.)
And most likely, transposing to another key won't help, because you won't be able to get all the intervals in the song's melody. Your instrument, like a standard harmonica or tin whistle, is built to produce only the notes of one or two diatonic scales, and your song is apparently more chromatic. Unlike the harmonica, "bending" notes isn't a viable option. Unlike the whistle, you can't cover half a hole to get the in-between notes.
So enjoy your concertina for what you can do with it, but if that particular song is a priority for you, you'll need a different instrument to play it on.
By the way, what song is it?
I'm in a Disney singing group, where we sing Disneyland attractions' songs. So naturally it would be Pirates of the Carribbean. I appreciate the responses I've received and it looks like I'm going to have to invest in the 30 key, to have the accidentals I need.