No. This is definitely not agile. Nor is it a good idea.
Cross-functional teams, i.e. teams that include every role (analyst, server admin, database admin, UX designer, QA tester, technical writer, graphic designer) that is needed to successfully deliver working software, are a staple of many agile methodologies. In fact, in many methodologies, the customer itself is also considered to be part of the team.
Actually, this is orthogonal to agile, though. Cross-functional teams are simply a good idea, regardless of whether you are doing agile or not.
What is true, however, is that with the constant rise of automated testing, developer testing, test-driven and behavior-driven development on the one hand, and software-defined infrastructure, highly automated commissioning, configuration, and deployment, DevOps, and cloud hosting, some of the workloads may have shifted from admins to DevOps engineers, and from QA to development. That does not, however, mean that those roles are extinct. It just means that QA has more interesting bugs to chase because all the trivial ones have been found by developer testing, and admins focus more on enabling DevOps to manage the infrastructure with automated tools than managing it themselves.
There is an easy test to check whether something is agile: when someone says "you do this because it is agile", then it is not agile. Agile is all about self-managing teams that constantly reflect on their processes and adapt them. Whenever someone says "you do this", then it is not agile. It is only agile when the team says "we do this, because after reflecting on our past experiences, we have determined that it works, and we will keep reflecting on it and stop doing it as soon as we determine it doesn't."