I think I might have found the answer (at the official CIC website) you did not want to hear:
Who is a federal skilled worker?
Skilled workers are chosen as permanent residents based on their education, work experience, knowledge of English and/or French, and other factors. These things often help them succeed in Canada.
People can apply to immigrate to Canada under "Federal skilled worker" category. Skilled workers apply under this category if they want to live in any Canadian province or territory except Quebec.
Skilled workers who want to live in Quebec apply under a separate category. The province of Quebec selects its own skilled workers. Applicants who intend to go to Quebec are known as "Quebec-selected" skilled workers.
source: http://www.cic.gc.ca/english/helpcentre/answer.asp?q=106&t=6
Given the above answer, for you to apply for FSW, you need to show that you do
not intend to immigrate to the province of Quebec.
If some how you can show/prove that your
intention is to go to a province outside of Quebec then you would be ok with FSW application. Once you get your FSW you can use it anywhere in Canada, including Quebec.
It is quite plausible that you
live in Ontario but you then got a job in Quebec.....and as such you are
relocating to work in Quebec......I don't know how you are gonna explain all of this to CIC though, but its POSSIBLE....good luck.
Also, note the below statement makes reference to "live", it did NOT say "work" and it did NOT say "live" and "work".......it ONLY said "live".
"Skilled workers apply under this category if they want to live in any Canadian province or territory except Quebec."
You are applying for FSW because you are going to "
live" in Ontario......there is no need for specific reference to where you are going to work....the regulations makes no such reference........so you could very well live in Ontario.....but work in Quebec.
(....I am not a lawyer though)