You add code to Calibre and the ereader software to communicate and store the current page. Or you pay someone to do that. Or you explain why this is desirable and that way make others want to make these changes for free.
It is already possible to store the current reading position between several devices, using an ereader that send and store the position to some file on the internet/the cloud. This might be enough? If you wish to publish or post this information, it might be easiest to do so directly from the modified ereader, without involving Calibre. Or Calibre could read the reading position from the cloud when needed. That way Calibre doesn't have to be running while you're reading.
If you can figure out exactly how some existing ereader store the reading position in the cloud, then you should be able to write a plugin to Calibre that can read this reading position.
Calibre is open sourced, so the code is available for updates by you or anyone else. If you choose an ereader that also is open sourced, then that code also is available. However it might be best if you can get the code included in the official code, that way you can keep updating Calibre and the ereader.