The below was prompted by the question "When was Thom and Moiraine supposed to happen?" This was a comment I am migrating here and removing from the source forum.
It was there all along, but subtly. Ready for the full breakdown?
The start of their journey together was a meeting of mutual respect in which both prominently displayed their conspiratorial natures to the other. Her hinting that she recognized his name as the Court Bard of Caemlyn (addressing him as Master Bard), him acknowledging her omission of the Sedai honorific. With only a couple words each they told one another "I know your secret, but I'll play along for now."
And then along their journey to Baerlon - while Lan scouted ahead, the Emonds Fielders lagged behind. Which leaves Thom and Moiraine in the background together. I don't think there was anything romantic here, but rather this is where their friendship started to develop.
What changed was when they arrived in Baerlon and Min revealed to Moiraine a Viewing that she would marry Thom. That night she danced to his music, and it was the only time she let her hair down. On the journey that ended with them split at Shadar Logoth, the same held true again. While Lan was scouting ahead, the Emond's Fielders were lagging behind, leaving Thom and Moiraine exploring their friendship offscreen - with Moiraine having embraced a romantic interest in Thom.
Being as both are spymasters with extensive practice controlling their responses to emotion, this likely only amounted to an inclination to talk more and ask more questions. Again, nothing overtly romantic.
But when they split at Shadar Logoth, Thom follows her lead, despite his natural distrust of Aes Sedai and his want to protect the kids from her. When they arrive in Whitebridge, he sticks to her plan, and initially intends to wait for her to find them. This is where you can see that he's overcome his distrust of Aes Sedai regarding her - that he has some degree of faith in her.
Later, when rediscovered, one of the first things he asks Rand is if she's with him. I think most people read that question as suspicious or distrustful of her. But think of it again in the context of the above, and everything that comes after. He was hoping to see her again.
When he starts the civil war in Cairhien and walked away, where did he go? Tar Valon. Why? To find Moiraine. How do we know? When Mat finds him, he's drunk, playing in a tavern, and all he wants to talk about is what a fine woman Moiraine is. So he goes with Mat, and over the course of events reunites with her in Tear.
And Tear is where they really started flirting. The scene where she's "rifling through his room" is full of that. She's talking to him about the things that matter to him - Owyn and the Vileness, and she offers to betray the White Tower for him. Beyond that, what she doesn't reveal to him, is that she also implies elsewhere that she uses one of her 3 questions to get the answers.
And they spent 4 months weeks playing the Game of Houses together and bouncing the Tairen nobility off each other. They were having fun together, reveling in doing the thing they loved most - intrigue. They found someone they could share that with, someone they felt comfortable taking the mask off for.
In his room, one of my favorite exchanges, is when she reminisces about meeting him in Emond's Field. She does so by teasing him about how he used his real name and expected her not to notice. Of course, this is a low key, "remember how I knew who you were when we first met?" It's a stroke to his ego while simultaneously flexing about her awareness, and it calls to a fond moment in which they first started to respect each other.
She says to Egwene that she knows the face of the man she'll marry (Thom) better than they know theirs. When she leaves him in Tear she promises they'll see each other again.
You can also see from their inner monologue that each thinks of the other as their first reference in cases where the other has any ability at all. Moiraine observes someone juggling IIRC and thinks it pales in comparison it to Thom's dexterity, for instance.
They weren't hanging from the parapets and boning, they probably never even kissed. I'd wager there were a couple moments neither followed through on, because they're both reserved and mission driven. Their friendship started almost immediately, and it built up gradually.
It's a case study of "you don't know what you've got til it's gone." After Tear, he wouldn't see her again until the Tower of Ghenjei. After receiving her letter, he agonized over not being there for her, straight up until they got out of the Tower, and even some time after. He had developed feelings for her, and his time in Tear with her solidified that, but he had denied them up until then and not acted on them after accepting it.
And she wasn't going to force it - she knew it would come in time. Her primary mission was always seeing Rand to the Last Battle. When she left him, she knew she'd see him again, but didn't know how bad shit was going to go for her until Rhuidean.