Quebec Infraction under my name that never took place, how to proceed?

I was living in Montreal between May and August of last year for an internship and have been back in my Ontario home city ever since.

While there I only had one ticket for jaywalking on June 30th, 2018; paid it off the next week.

1 hour ago I got a letter in the mail from SAA saying my Quebec Driver's License will be suspended due to an unpaid fine (funny because I only have an Ontario License and never drove a car in Montreal).

I called the phone line and then Le cours municipale and was told on July 4th I got a ticket for riding a Bike without headphones?

I got a warning from the police when that new law came into effect on a totally different date (early August) but there was never an exchange of name or ID as it was a casual friendly reminder/warning and thereafter I stopped wearing headphones because after my jaywalking incident I learned Montreal Cops are sticklers on non-Francophones and a $150+fees ticket was not something I wanted especially because I only had 1 month left in the city when I got the warning and I'm a broke student

I explained the situation to the phone agent and asked for information such as "what address was provided, what type of ID was provided, what name was provided?"

And was told that only the general address to a 200 unit apartment building (that I did live in) was given, no ID was given, and my middle name was absent from the file.

Vs for my jaywalking ticket the correct address was on file, my full name with middle name and my Ontario License # was on record.

I think it's ridiculous that I'm getting a fine in the mail 1 year later when I was never even given notice that someone evidently gave my name to the police and that the police actually went ahead and processed it with no proper ID, no full name, and no full address...

I tried telling the agent that just off the fact that my full name is not on record, that is already a fatal error and the fine should be cancelled, that no ID was given, no address, etc etc

And she told me I have to go to Montreal to explain it in front of a judge, that's absolutely insane that I am expected to drive from Toronto to Montreal over a $150 fine that is illegal and that I never actually received (and which the lack of full information in the file supports)

I really loved Montreal and wanted to move there after I finish school, but [...] the law enforcement and bureaucracy is such a turn off

Anyone have any advice?