Critical, show-stopping bugs are one thing. You should always try to test for those as much as possible. And there's will always be a slim chance that some really weird edge case will occur. Best way to handle that is patch it up, create a test for it, and move on.
However, I think it's practically impossible to do what they are demanding. Most important thing to me is that there's no definition of what is considered a "bug."
Is there a styling that they don't like the look of? Is the user interface not exactly doing what they expected?
That's when you need to really examine: is this a bug or an unintended outcome? For example, if I made the text blue with no requirement whatsoever saying that it had to be blue or not be red or be any color, then the project manager yells at me because it looks better in green (and there was no requirement for green), I would argue that that is not a bug and that I couldn't have been expected to know that. Make sure when they say it's a bug: it is something that is a substantially unexpected behavior when something was explicitly expected.
Now I'm not saying that you need to get really nitpicky with what is and isn't a bug. The last thing you want is to start butting heads with your project manager over such a thing. What you want to do is get on the same page on things: acknowledge that you agree that there should ideally be zero bugs, but you cannot go in expecting that it will never ever happen. It's impossible to know the infinite combinations of actions and expectations to create all the tests for them. And bugs will almost certainly happen in many different degrees. What's important is that you address them professionally and timely while minimizing user impact.
Also, maybe unrelated to the main question, but I think could be helpful: when I have to deal with someone (a manager, client, user, whatever) that is angry at me: I acknowledge their concerns, focus on figuring out what what their problem is, and respond with empathy and logic. Determine what they are upset about, do (or in some cases not do) what needs to be done to resolve it, then talk about it. Stay calm, think, and respond. Talking and communicating are often overlooked and needed when both parties need to trust in each other to get things done.