No. Lack of a central coherence violates a core principle of the Scrum framework. Don't do this by choice.
Sequential Work is Usually Evidence of Organizational Dysfunction
The team is thinking to have sprint 1 and sprint 2 (with four developers) start on the same date, but sprint 2 to have 3 weeks cycle to accommodate the QA testing
Testing post-facto is a well-known agile anti-pattern. This is generally done to avoid dealing with the lack of a cross-functional team upfront. In other words, it's an attempt to bury an organizational dysfunction behind something that sounds agile-ish, like "parallel Sprints."
For Scrum, the rule of thumb is:
One Product ➡️ One Project ➡️ One Team ➡️ One Integrated Increment
Even in scaled agile implementations, you must still have an integrated increment each iteration, even if you have multiple teams. That isn't the case here, though; there's no way to create an integrated increment with teams split along functional lines the way you propose.
Integrate Your Teams with Test-First Development
A core agile practice is to do test-first development. By integrating your testers and developers into a single team, you ensure that the team as a whole maintains a central coherence (which is required by the Scrum framework). You will also generally increase product quality and reduce technical debt over the life of the project by leveraging collaborative test-first practices.
While there are certainly cases where you can't fix an organizational dysfunction, the job of the Scrum Master is to ensure that:
- the dysfunction is surfaced
- acknowledged by the whole Scrum Team
- accepted as a business risk by senior management, and
- kept visible as an ongoing cost to the project.
If you don't do those things, you aren't really following agile principles and best practices. As a result, you are unlikely to reap the benefits that a truly agile implementation would provide.
If management won't allow you to integrate the testers and developers, they are creating project risk. Management always owns business risk, so if they break the process they get to keep both halves.