Steve,

It would totally depend on the phone system or if the phone system has a compatible Call Manager or Call Assist program. For example, Panasonic Business Phone systems have a Call Manager add-on by Poltys - Poltys Assistant, according to their site, it integrates with Windows-based CRM programs.
https://pbx.poltys.com/en/products/Assistant/poltys_assistant_overview.html

As a Panasonic Dealer, I can tell you I have not yet tried this software out, so i cannot tell you if it will integrate with Adagio, but you could contact them & see?

Good Luck