Unlike the others who have responded above... for me, absolutely they can notice it.
It seems to me that I treat the definition of "Perception" differently than other people (and probably how the game treats it too for the most part.) To me, Perception is noticing things. NOT "looking" for things, or "searching" for things-- in my games, that's why we have Investigation as a skill.
In my games, Investigation is used for deduction based on information you already have. For example, you know where the secret door or trap is and now you're trying to figure out how it works so you can open it or, in the case of traps, avoid or disarm it safely. It might also be used to deduce a monster's abilities, strengths, or weaknesses based on observing it. There are many other applications, but essentially it boils down to the character deducing something.
I don't go along with the other people who commented that if a PC didn't say they were searching for a secret door that they had no chance to notice it. To me, that's ridiculous. What if the secret was just hidden really badly?
I believe a secret door that is hidden really badly is just called a "door."
Here's an example of how I handle things from a one-shot that I'm running Wednesday night:
So as they move through the dungeon, each character is undertaking a particular task of their choice. Each of these tasks has by default an uncertain outcome. If you're doing any given task, you're not doing another task, so there's a meaningful choice to be made there. Your pace also determines the DC. The slower you go, the lower the DC, and the higher the chance of a wandering monster happening upon you. The faster you go, the higher the DC, and the lower the chance of a wandering monster happening upon you.
If you want to find all the secret doors, for example, you have to dedicate at least one party member to that task who has a passive Perception of at least 15 and move at a Slow pace. Or put someone with a PP 10 in there and have another PC Work Together with him or her. Or you can move at a Normal pace if someone has PP 15 and someone Working Together with him or her. The PC tasked with finding the secret doors is automatically surprised if monsters are being stealthy as is the person Working Together. That's the risk for finding all those secret doors.