Are you using ZA Pro or free ?
ashQuick.exe (right click context menu, quick file scanner) doesn't require internet access, the only ones that do are avast.setup (the avast update process), ashMaiSv.exe (Internet Mail scanner) and ashWebSv.exe (Web Shield).
I haven't used ZA for more than three years but, my Outpost Pro has recently been updated to version 4.0 and I notice more pop-ups about avast processes, like ashDisp.exe (avast system tray icon) calling/initiating say an avast update (avast.setup), when I initiate a manual update. But, it isn't connecting to the internet and since it was a valid operation as I was running a manual update then I allowed it.
Perhaps this is some form of tightening outbound security (leak testing/checking), in ZA as it is in Outpost.