Yeah, you can install QSE on multiple servers which can be either on-prem and/or hosted by a cloud provider. The only thing you will need to do is make sure that the servers can talk to one another, which is less so a Qlik thing more so a SysAdmin responsibility. To the machines, it won't matter whether it is in the cloud or in your company's data center.

With that said, "elastic" is a whole other topic. If you want to dynamically scale, without the use of kubernetes, you will have to manage that all on your own. That means you will need to build the scripts to spin up a node in the cloud when usage hits a threshold. You'll need a way to spin down a node when usage is low, and unobtrusively move the users remaining (probably the hardest part) from the node about to be spin down. You'd also need to have an external load balancer which routes traffic accordingly. This is definitely not a small feat, but doable.