Well, it could be a really sweet moment. Maybe it happens after they've been through a tough case together. They've been growing closer and closer, and one day, after sharing a deep conversation, their eyes meet and they just can't help but kiss. It would be a moment full of unspoken feelings finally being expressed.
Perhaps in the fanfiction, they are training together for a new police technique. They keep messing up and laughing at each other. But then, in a moment of stillness, they realize how much they enjoy each other's company, and the first kiss is a gentle, spontaneous expression of that new - found feeling.
Yes, there are quite a few. Some fanfictions might explore their relationship further after the kiss, delving into their emotions and how it changes their dynamic at work and in their personal lives.
In many fanfictions, Natsu and Lucy's first kiss is often a very emotional and anticipated moment. Some writers might describe it as a spontaneous act during a dangerous adventure, like when they think they might not survive. Natsu, being his impulsive self, might suddenly realize his deep feelings for Lucy and plant a kiss on her lips.
It deepens their relationship immediately. After the first kiss, they become more aware of their feelings for each other. There's a new level of closeness and they start to rely on each other more emotionally. They might start to hold hands more often, and share more private conversations.
You can start by looking on popular fanfiction websites like Archive of Our Own. It has a large collection of fanfictions related to various shows, including 'The Rookie', and you might find some great ones about Tim and Lucy's kiss there.
I'm not sure specifically as you haven't given much detail about it. It could be a story about their friendship, a love story between them, or an adventure they had together.