From what I understand you need to write the who, what, where, why and how of testing within your organisation. That is what testing frameworks usually detail.
You can take a look at the development life cycle that your organisation uses. If you get the specs and code after they’re complete then you likely have a waterfall methodology. The v-model is a variation on the waterfall to increase testing input. I won't talk about the pros and cons here of the various models as you will need to see what best fits your organisation.
When it comes to the framework; most of the ones I've seen, read or written have been leveraged from existing ones and tailored to suite.
What you need to define is who is doing the work, how the development and business analysis teams interact with the testing team. What is tested? What testing processes are used by the testing team, what the testing timeframes and timelines are. Where testing is undertaken. What environments exist and the entry-exit criteria for those environments. Why testing is done and what it is meant to achieve within the context of your organisation. The goals of testing are different for a webdev shop than a defence organisation.
On top of all that you'll need to document the how. How your team tests. What the process is, what is documented and where it stored. What metrics are gathered during the testing cycles and to who gets visibility into those metrics.
You can take a look at the IEEE standard for software testing, it's 829 I think. Personally I find it heavyweight and outdated but it may be relevant or at least a great starting point. In any case the final framework has to be tailored to your organisation and how your org works.
When you write this document you should engage the developers and analysts to contribute as well as including feedback from your team. It is for everyone. Your testers will use it for direction and the other teams for test engagement.
So the short answer is IEEE829 or whatever as a PRINCE2 equivalent but in the end I believe you need to ask the questions above about your testing setup and tailor the answers for how you and your team work.