No, I don't have the icon, but that's the icon that was in the pop up message! Phew! I'm so glad I did not agree to install anything! Thanks Freepers for confirming that I was right, and thanks for all the advice.
I’m not an expert in these things, so someone from my employer’s desktop support team spent about an hour going through various folders and registry files deleting stuff that had been installed by this program. The problem is now solved. Here’s the one thing I was able to do myself. Based on some web research done by my wife using our non-infected computer, we learned that the malware I had was called “Antivirus Live”. We learned how to temporarily disable the program. I opened the task manager and looked for a process called ***sysrequest.exe (where the *** are random characters). When I spotted that process running, I knew for sure that my problem was “Antivirus Live”. At that point, I selected that process and stopped it. That ended the annoying program and stopped the popups for that session. If I had rebooted at that point, the program would have started itself again, so it was at that point I brought in the experts to remove it. Incidentally, I also learned that most antivirus software doesn’t detect this particular program. Running a typical virus scan probably won’t detect or solve the problem.
I should add that my problem went deeper than the current browser session. I had already shut everything down and rebooted my machine. The popups were still there, even when I wasn’t connected to the Web, so I knew something had been installed on my machine. You may not actually be infected yet. See if there’s still a problem after you reboot the machine.