Event One

Medic and Woodsman first met when Woodsman had to go through the  operation of getting the ubercharge heart implant, but as Medic does, he  also gave Woodsman his top surgery. Honestly Woodsman was kind of  scared of Medic at first, and would purposely avoid check ups from him  just so he wouldn't have to see him.
There was no hard feelings over Woodsmans fear exactly, he was honestly just scared of Medics sudden and easy welcomeness and enthusiasm, being awake during the operation might have helped with that, that kind of freaked Woodsman out.

Event Two

Woodsman started to open up to Medic more when Medic would tell stories about Engie during the time Engineer was gone every time Woody had to go into Medics office. Medic would talk for hours about things that the mercs have done and stupid stories of everyone there as Medics way of making everyone seem less intimidating. Medic didn't ever notice that Woody was ever scared of him, and was kind of oblivious to the whole thing and was too busy talking.
Once Woody started to open up more and feel more easy around Medic, he would often go into his office at random times just so Medic would talk to him and tell him funny and random stories he had, whether it be about the other mercs or funny stories about his past. In return, Medic liked hearing stories Woodsman had about his past with Engineer.

Event Three

Medic and Woodsman later became very good partners on the field. The other mercs of course started to notice how close Woody and Medic have become, and basically tease them by telling people they're inseparable, mainly Sniper and Spy.
Medic was the one who found out that Woodsman liked him. He found a note or letter of some kind in his room when Engineer once asked him to grab something off of his desk, and since Engineer and Woodsman share a room, Medic had accidentally stumbled across it on the floor, in which it looked like Woodsman was trying to throw it away.
At the time, Woodsman was out doing a mission so Medic didn't confront Woody immediately, but he did ask Engineer about it. Engineer talked to Medic about Woodsman and how he just wanted to see him happy, and to just try to be careful with him, even though he knew he had completely good intentions, Engineer had seen them interact a lot after all.
It took a few days to finally confront Woody about it, mainly because Medic wanted to think of a good, nonthreatening way to approach the situation, and got advice from Engineer of course. Once Medic finally talked to Woodsman about it and they told each other how they felt, it took a few days for them to make it public to the other mercs, but neither of them told the others they were officially dating, Medic just suddenly started showering Woody in affection and the rest of them just had to put two and two together.