Quite a few people have requested something like this, but Chrome doesn't allow to open popup without user action (it used to be possible quite some time ago, but doesn't work anymore).
What I can do instead is that I can have a desktop notification appear if there are notes for a site. However be aware that chrome desktop notifications can be a tad annoying if you have notes for a lot of webpages and I am not sure that's what you want. I can update the page notes testing version
to include desktop notifications; will you guys be willing to give it a try?