Currently HPE ALM Octane is only being delivered as a ''Cloud Based Service'' through HPE SAAS. Octane is not the same as HPE ALM 12.53, HPE ALM 12.53 (or ALM.net as it is sometimes referred to) is the Quality Center type platform that you are currently using. HPE ALM Octane is a new product that is meant to integrate with ''ALM.net'' in order to provide additional features (mostly related to Agile and CI Servers) it is not meant to replace the current ALM platform.
There is not any direct upgrade path to switch to ALM Octane, if you wanted to use features available in ALM 12.53 you would still need to upgrade your ALM server from 12.20 to 12.53.
Eventually HPE ALM Octane will be able to installed locally but for right now it can be used ''in the cloud'' here:
https://saas.hpe.com/en-us/software/alm-octane
Since it has not been released to install locally it would be hard to say if it would support older versions (such as 12.20) in that configuration, I would assume that your ALM version would need to be at the same version as Octane but we will have to wait for the ''full release'' to find out if this is the case.
For right now you can access the saas version above and begin to explore the features, in the future it will be available for local downloads/installations and we will know more about what versions it will work with.