No, at least not in the United States. We never apply the adjective "punctured" to the bicycle.
We can say "I got a punctured tire," or "I got a puncture," or "my tire was punctured."
We would be more likely to say "I got a flat tire," "my tire went flat," or "I got a flat." Strangely, we can't say "my bicycle got a puncture" but we _can_ say "my bicycle got a flat."
"Flat" is less precise than "puncture" because it doesn't say why the tire went flat, but it's probably what we'd say. We'd be more likely to use the word "puncture" if we were going to go on and talk about what made the puncture. "Be careful on Pleasant Street today. I don't know what happened but there are a lot of nails in the road near the intersection with Winslow, and one of them punctured my tire."