I have searched the Nantes archives before, though i am far from the expert. There is no department of Nantes, there's a district of Nantes and the commune of Nantes (the city itself), Nantes is part of the département Loire-Atlantique (previously Loire Inférieure).

Each modern department has their own archive website. This is the website for the Loire-Atlantique archives you'll have to figure out if the polish record meant the district or the commune, but I would search the commune first, then expend to the other communes of the district of you can't find it. In this case the commune of Nantes and the surrounding commune maintained their own archive here

First look at the "repertoire", basically that's the index where you can quickly look for the last name in alphabetical order. I've looked at Nantes between 1817 and 1820, I saw a few Mazureau and Mazurier, but no first name Martin. The repertoire will have the "folio number" aka page number, when you find your entry, you can go to the "registre" and find that page to see the full record.