I always think I sound pretty anglo-Canadian when I speak English, but people outside of Québec tell me my English 'sounds French', and people in Québec tell me that indeed my English does 'sound English'. I still consider myself pretty accent-less when it comes to speaking English- any remnants of my old accent kind of faded when I moved to Canada, and I moved here young enough that I hardly had an accent to begin with.
When I speak French I sound Québécois. I've grown accustomed to far too much local slang/manner of speaking that my French is nearly undecipherable at times to someone who have their French from elsewhere. I try to tone it down because people really do get confused, lol. Words get very smashed and slurred together.
French really influences vocabulary when speaking English.
For example, a general store or a corner store is called a dep, a newspaper is called a journal, a notebook is called a cahier, a combo meal (burger with fries or something) is a trio, 'fries with gravy and cheese' is poutine, a winter hat is a tuque (I think that's common throughout Canada?), the subway/underground is called the métro, etc. Of course, an English-speaker here is 'an anglo'. Some sentence structure also gets a little muddled up with people who switch between French and English a lot.
Because of langauge laws, store names are also different here- KFC is PFK, Shopper's Drug Mart is Pharmaprix (learnt that on a recent trip to Ottawa!), Staples is Bureau en Gros. Those are the most well-known examples.
So, even if you're an anglo in Québec, French still changes how you speak English. Even my most anglo friends here find themselves occasionally switching into moments of French.