Should we have separate development & testing sprint in Scrum

My team has been raising this issue again and again that they do not get sufficient time to test as in 2 weeks sprint PBI is delivered on 5th day and QA has to complete the testing in remaining days. There are chances that they could find bug at last moment which would eventually lead to spill over.

Why doesn't the team reduce its work in progress limits, so Product Backlog items are available for test early and often? Focus is essential to Scrum.

So they wanted to have development done in Sprint 1 & testing in Sprint 2 and so on which will ease out the pressure. 

The Scrum Guide says "Sprints are the heartbeat of Scrum, where ideas are turned into value." The proposal being made would result in two fake Sprints, neither of which would succeed in these terms. It might feel as if pressure is being eased out because it would cover the problem up.

If you split the 2 Sprints as you said the next complaints you will hear is that the Developers do not have enough work to do during the Test Sprint and the Testers do not have enough work to do during the Development Sprint. 

That is not fixing the problem you have.  It is trying to hide it.  The problem is that Scrum Team has sub-teams within the Developers.  

The section of the Scrum Guide that explains the Scrum Team opens with this paragraph

The fundamental unit of Scrum is a small team of people, a Scrum Team. The Scrum Team consists of one Scrum Master, one Product Owner, and Developers. Within a Scrum Team, there are no sub-teams or hierarchies. It is a cohesive unit of professionals focused on one objective at a time, the Product Goal.

Developers are defined as 

Developers are the people in the Scrum Team that are committed to creating any aspect of a usable Increment each Sprint.

So, your QA would be considered Developers in the Scrum responsibility definition.  I suggest that you encourage them to investigate better agile testing methods.  I have a 20+ year background in QA.  I find it hard to believe that 5 days of development takes 5 dedicated days to test.  Especially in a Scrum environment.  There should be potential to test things from the first day of the Sprint if you are breaking your Product Backlog into small incremental units of work.  Your description sounds very much like you are doing waterfall process in 2 week increments.  Just saying that your are "doing sprints" does not mean you are being agile in what you do.