Oddly, I don't know. Honestly I think it's totally impossible, even in principle, for nothing in the universe to be moving/propagating because everything in the universe is, fundamentally, a process. But you could ask, what if, hypothetically, things could stop moving? Would time exist then? And that gets tricky. That hypothetical universe wouldn't be ours, so you could say it depends no what universe you're talking about. Or, you could say that whether time would exist without motion is independent of what universe you're in and is a basic logical or metaphysical question. But then you could say that maybe there's no possible universe in which things could exist as processes and then suddenly stop. Or you could even say that there's no possible universe in which things exist not as processes at all to begin with, but I'm not sure if that changes the answer to the question, and I'm not sure if it's true either. And you could even say that the fact that there's no possible universe in which things could just stop even in principle doesn't stop us from logically hypothesizing about whether time would stop. I guess in that case it's an epistemic/semantic question of what "time" means. But I guess the question should go deeper than semantics: what is time, ontologically? I think it's safe to say that without anything in the universe, there would be no time.

Actually, here's a good question:

If nothing in the universe is moving and it stops time, for how long would it not be moving? The answer becomes meaningless. Which means it can never start moving again. Or, put another way, time and movement and propagation already "stop" for an infinite amount of "time" every nanosecond, but there's no way we could notice it. In other words, time necessarily advances at a rate of "1 second per second."