Interesting! I don't know LSQ or French, but I used Google to find some information:
Have you tried the site SignPuddle? It shows LSQ signs using signwriting. You have to know French to look up signs in LSQ. And you need to know signwriting to understand how to sign the words you look up. I learned ASL from my parents and other people, not through signwriting or books. But if you are learning ASL in a class, then you might be able to use signwriting.
Here's the SignPuddle page for the sign for "chat" (cat): clicky!
and the page for "chien" (dog): clicky!
Those are the only two words I know in French, so I can't look up more. But there are online English/French dictionaries that you could use to get more French words and find the LSQ sign.
Here's a comparison of ASL and SL from other countries: clicky! It doesn't say much, but it is interesting.
I also found this quote on a page for (I think hearing) people who teach ASL: "As you know, ASL was based on the sign language in use by the National Royal Institute for the Deaf in France and today, standard ASL bears clear similarities to French. You may also know that in Québec, Deaf use a variant of ASL called LSQ (Langue Signé du Québec). I stress that LSQ is not a different sign language, it is ASL but modified to suit the cultural and linguistic needs of a French speaking environment." I didn't know ASL and French were related. Does this mean I can speak French now? 
