I am sure you can sponsor your wife to Canada, and that you don't have to sponsor her through Quebec.
Look at form 5540, the Sponsor Questionnaire, question 11: Are you currently living in Canada?
If 'yes', they want your address.
If 'no', they want the date you will return, where you plan to live in Canada, and proof of your plans to reestablish in Canada.
I am Canadian, but staying (temporarily) in China with my wife while we wait for her visa. I put our China address as my address (and will give my sister's address as our temporary Canadian address when my wife lands). I outlined our plans to return to Canada -- but there are very-good other posts on how to detail plans to return to Canada, so look for them elsewhere.
Since using your friend's address in Ontario is out,
I missed that part; why is the friend's address out?
On a separate sheet of paper (staple it to the form, and label it as further explanation for question 11)
By all means label the extra paper, but I believe the instruction expressly forbid stapling.
Then provide all the proof of this you can get. A paid-up apartment lease in Ontario would be great - your friend could provide one, couldn't he, even if he doesn't want you using the apartment as a mailing address now? That letter you mentioned he could write would be good. A firm job offer, if possible; if not, evidence of a job search. You, your wife, or your child could apply to some type of school (university, English classes, daycare) in Toronto, and if possible pay for it - a letter of acceptance would be good evidence; a receipt for classes even better. A letter or affidavit from your parents stating that they know of your plans to move to Ontario would be good. Further explain that you and your child are just visiting your wife in Ecuador right now. What kind of visa are you on? If it is a temporary visa or a visiting visa, great, show them.
All this is good advice.
I think you will find out quite quickly if this will work or not. Probably when they review the application in Mississauga before sending it to Bogota, they will notice you are residing in Quebec but are not applying through Quebec. You won't have the required forms for Quebec, so if they don't accept this situation, they will probably send it back then.
I'd keep the story as simple as possible, to avoid Mississauga's sending it back. If your friend won't allow you to use his address, then outline your other plans, letter from friends or family welcoming you to stay with them (this may require a few bottles of fine scotch), etc. My sponsorship application was approved in the strength of my written plans ( a few paragraphs, no more), and a signed lease (by now I have forfeited the deposit, since Hong Kong is taking longer than I wished to process the application).