Hello candianscottish1985,
I'm not an expert on spousal sponsorship, but your question was interesting so I did a little bit of research and here are my 2 cents:
The essential question that will be asked by CIC is: "Where does
the sponsor live?". CIC's website has this to say about the eligibility of the
sponsor:
If you live in Quebec, you must also meet Quebec’s conditions to be a sponsor
http://www.cic.gc.ca/english/immigrate/sponsor/spouse.asp
Similarly, Quebec's immigration website has this to say about the sponsor:
You can sponsor a close relative if you are a Canadian citizen or permanent resident living in Québec
http://www.immigration-quebec.gouv.qc.ca/en/immigrate-settle/sponsors-sponsored/requirements-sponsor/index.html
Note that nowhere is anything mentioned about where the applicant, i.e., you, lives or plans to live. However, I think it is safe to assume that any government agency expects you (the applicant) to follow and reside with your Canadian partner (the sponsor).
Therefore, if you girlfriend is also moving to QC with you, I would expect that once she updates her address with CIC, that they will request her to go through the extra QC step and get a CSQ for you. This will add some cost and time (a few weeks?) to the process but, IMO if you are indeed planning to settle in QC, I believe the CSQ will provide you with some perks (access to free French classes, QC healthcare without having to go through Ontario's first, etc.) and with peace of mind (avoid any potential claims of misrepresentation: you planned to settle in QC, so you applied through the QC process. Period.)
If you decide to follow that route, and your girlfriend hasn't been approved as a sponsor yet, I believe she should just change her address with CIC and file any extra forms, if applicable. Once approved (as sponsor), she should apply with MIDI for their part of the process to get you a CSQ, which will then be used for your part of the process (applying for the PR status).
If, on the other hand, your girlfriend is staying in ON, or anywhere elsewhere in Canada for that matter, she won't need to (and probably can't) go through the QC process (since it all hinges on where the sponsor resides). However, this also means that you won't get a CSQ and you will have to "land" anywhere but QC and "later" move there. I believe you also wouldn't be eligible for integration/francisation services upon landing**, and if you go apply for your health card as a new immigrant without CSQ this may raise some serious questions. The way I see it, even though you'd be following all the rules applicable to your situation, it would still look bad (you'd have to pretend, or at least imply, that you're settling somewhere other than where you really are). It seems that the system isn’t designed with this case in mind, but some could argue that the point of spousal sponsorship is for both partners to be together and that in fact filing an application where each person lives in a different province, and it doesn’t seem to be a temporary situation, may raise questions about the authenticity of the relationship to begin with.
My opinion: make up your mind about where
the two of you will settle (QC vs not-QC), and if that’s going to be QC, then get an address there as soon as possible and have your girlfriend apply for the QC sponsorship part of the process.
Relevant resources:
1. CIC operational manuals IP-2:
http://www.cic.gc.ca/english/resources/manuals/ip/ip02-eng.pdf
2. CIC operational manuals IP-8:
http://www.cic.gc.ca/english/resources/manuals/ip/ip08-eng.pdf
3. Quebec’s Guide des Procédures d’Immigration (regroupement familial), in french :
http://www.midi.gouv.qc.ca/fr/publications/guide-procedures-immigration/regroupement-familial.html
** I am not 100% sure about this, but I'm sure you can inquire about that from the Quebec government directly, if interested. Specifically, what exactly is the added value a CSQ provides when a PR decides to move to Quebec. And is there any other way someone without a CSQ could receive the same perks, if any? 
