So I'm trying to work out how an equivalent to a perception check would function. Would I just have them roll for logic? Should I add negative dice for there being no danger while just trying to notice something off about a survivor or anything?

What are your thoughts?

I would just more or less try to tell them what they are aware of rather than make them roll a perception check. Unlike d&d where I call perception checks all the time, I'd just try to tell the PCs what they can see or hear or smell at all times, clarifying when they ask. I can see certain situations though were a further roll might be called for and I think logic would probably work. For noticing something off about a survivor I'd say that more of a social roll, using empathy.

And you can run tasks that have no chance of causing stress if there is no immediate danger (pg44).

I have my players roll logic tests from time to time. Its usually just 1 positive dice, if they succeed then they "notice that zombie quietly creeping up behind them"

but yeah, to see if a survivor is lying or something, then thats a empathy check.

In the session I played recently, my GM called for me to make a few logic-based perception tests. This is mostly, however, because I took the "inattentive" negative feature to represent how I often get lost on thought or focused on what I care about, usually resulting in me not really noticing the wider world.

So he'd have me roll 1 or 2 positive dice, and 1 negative for the inattentive (no stress, just the random chance of canceling due to me being stupid). This led to me missing a few news broadcasts and some sirens, but so far nothing major.

"Logic (offensive): Logic is your awareness of your surroundings and ability to think on your feet. If you are hacking into a secure computer system or noticing an ambush, Logic is the characteristic you are using."

So, yeah, it's Logic.