Test quality

There is no specific Quality Control team in my organisation unlike the big giants who do have dedicated support in this area. 

In my situation, I am dealing with a third-party supplier, a start-up organisation who are developing a software system. As a technical lead on the project, I am managing the test environment in Spira Team. As I have not witnessed any quality control mechanisms in place from the supplier end, I want to define a quality dimension to the project to evaluate the system quality and the test quality. I am seeking advice if there are ways of implementing the quality dimension within Spira Team and any other best practices you use to evaluate these parameters?

Thanks for your time in advance.