04 Honda Civic- Brakes make a pop sound when pushed

First of all what leads you to believe that there is no ABS on the car. Has to be. The pop noise; when you go to stop do you feel and shimmy on the steering wheel? I would certainly pull the front tires off and inspect the brake system. Look at the rotor for wear and the pads for how much life they have left.

There is ABS, my mistake. I thought the guy told us it didn't have it when we bought it, but I just checked to see if it was standard.
The steering wheel doesn't shimmy at all, the brakes work really well. There is just that pop when you push down on them, nothing else.
Thanks for your reply!

When you hear the pop sound do you feel anything like a jerk, vibration? I would be checking out the brakes looking for anything loose. A loose bolt, brake pads that are sloppy or loose.

No, we don't feel anything like that, just the pop underneath our feet. It's not a loud pop either, you would have to be looking for it to hear it from the passenger side.

Does the sound happen if you are not moving? Can you get the pop while sitting in the driveway?

I tried that just now. I put my foot on the brake just before I started the engine and it popped. I ran it for a few minutes, but I couldn't get it to do it again.

Look under the hood and see if there is a round rather flat container behind the brake master cylinder. I would call it a booster but that could be the source of the popping sound. Not normal for it to do that. Check to see that the hose to it and anything connected to it connected, not worn or loose.

We looked at everything connected to the booster, it looks fine. We both don't know much about brake systems, so we're going to have someone else who does take a look at it like you suggested. I'll probably be back here getting your opinion of what they say! Thanks so much for the help!