I have an HP C4385 Photosmart printer on my home network. I also have HP Solution Center and the relevant HP printer drivers installed on my computer. Whenever certain events happen on the printer (e.g. out of paper, cartridge alignment, cover open), a push notification appears on my computer. It looks something like this: 
Is there a way to disable these notifications? Edit: I tried following the instructions here: https://support.hp.com/ca-en/document/c01841764 But when I went to the Device Settings for my printer, it just looks like this, and I can't find the settings for printer status notifications: 
Also, I don't have seem to have any HP UPD item in my Devices and Printers page. 
The answer from @John only applies to printers that support the SNMP protocol. That only exists on network-connected printers.
It may be possible to achieve what you want. Check through the installed software and see what kind of utilities are installed. There probably is some installed package like HP Solution Centre . In that case, you may be able to achieve it, by doing something like these instructions for the "HP Utility" for a different printer model.
HP really should do something about their bloatware. Your problem is caused by the fact that for your printer (and many of their low-cost printers) the only available driver is the "Full Feature Software and Driver". I have looked for a "printer-only" driver, which would avoid this problem, but have not been able to find one. And scanning could then be done with a simple TWAIN driver - if HP supplied one.
