The quote doesn't end in a semicolon. The sentence contains a semicolon which appears immediately following a quote.
I'm having trouble imagining that, too.
But either way, treat it like other punctuation- which differs from place to place. In the US you would write: Good Sam asked, "Should I put it in the quotation marks?" In the UK you would write: Good Sam asked, "Should I put it in the quotation marks"?