Does "New Brunswick" Mean Scotland?

No. "New Brunswick" and "Scotland" are entirely different entities:

• New Brunswick – A Canadian province on the Atlantic coast (capital: Fredericton; largest city: Moncton). Named 1784 for George III, Duke of Brunswick-Lüneburg (German duchy).
• Scotland – A country within the United Kingdom (capital: Edinburgh). Name from Latin Scotia "land of the Scoti" (Gaels).

The only connection: both were named by British monarchs of the House of Hanover. No linguistic, geographic, or historical equivalence.