Similar to others a quality strategy is a very different thing from a Test strategy and it often greys into a quality action plan.
Strategy, get stake holders together and discuss quality, their goals and missions around quality and put in place an plan to manage quality throughout a project.
Picking up on another comment, our strategy could be to get stakeholders involved and gain to a good understanding of the available resources and limitations - people, technology, tools, schedules, deliverables, scope, documentation, testability, hardware - our mission and everything that supports it, ideally and create an action plan around those and then execute the plan.
Quality management plans or often simply quality plans may be what they are looking for. Usually not the testers responsibility but I dual hat and do these occasionally.
I had the following in a recent plan.
Quality goals and mission statement: Where are we now, short term and medium term business goals, any KPI’s or OKR’s to track quality against, primary stake holder needs, key quality milestones.
A set of principles that guide the plan.
Roles and responsibilities.
QA process activities, reviews, development model, retros, feedback loops, learning actions.
Change management process actions.
Environments, dev, CI process.
Development quality practices, reviews, unit tests, regression coverage, wc3 guidelines.
Acceptance criteria process.
Product quality risk register. Primary input into Test planning.
Lessons learned and improvement activities.
Documents and deliverables of quality information.
QA review log.
So basically it was a list of quality related activities and how we will leverage from them.
Now if it was a Test strategy then I’d also go for a variation of HTSM but that is a very different thing than a quality strategy or plan.