Conflict don't necessarily have to involve people; they can exist between ideas, situations and perceptions. Similarly, an antagonist doesn't have to be a specific character but instead can be things that block the protagonist from his/her desired goal.
In your case, the central conflict is between Lucy's perception (she thinks the neighbor is a spy) and reality (he's actually a nice guy). The "antagonist" therefore is not her neighbor but all the things that prevent her from "outing" him along the way.
In your description, you say her quest "never goes that well," so you already have failures and frustration in your plot. I suggest writing them in such a way that Lucy is convinced the neighbor is foiling her plans intentionally, thus driving her ambitions with greater intensity.
In this way, you build upon the central conflict and add depth to your story. The more convinced Lucy becomes that the neighbor is her antagonist, the greater the real conflict between her misconceptions and reality is emphasized. In the end, the realization that it was all "in her head" becomes the story's "moment of realization."
The realization that she was wrong all along might seem like a defeat, but it can be turned to a "victory" by revealing the neighbor is not only a "normal" guy but also one who encourages Lucy's imagination and energetic spirit (perhaps he is an inventor, or teacher, or artist). In the end, the antagonist (her misconceptions) is dispelled, but those things that defined her as protagonist (her imagination, curiosity and determination) are affirmed.