Like the previous two sections before this, this has nothing to do with what Quatre and Trowa's relationship categorically. Now, by the middle to the end of the series, it is undoubted that Trowa trusts and believes in Quatre. It is the beginning of their relationship that could be considered particularly strange. I would like to address that.
There have been plenty optimistic interpretations of their first encounter at the Corsica Base via fan-fiction and essays. I, for one, never really saw it in the light of "love at first sight". Rather, the entire time I saw Trowa as being on guard and wary of Quatre.
Yes, Quatre stopped the physical struggle and then vocally called out that they shouldn't be fighting. He then showed complete goodwill by opening the hatch to his Gundam's cockpit, rushing out and making himself completely vulnerable. Despite this, Trowa came out slowly with his arms raised in surrender. Even with Quatre's charming smile and friendly, yet amused comment about having surrendered first, Trowa remained unmoved.
We never do see what happened after that, but we do learn that Quatre brought Trowa back with him to the Maganac Headquarters: a lovely villa somewhere in the middle of the desert with plenty of room underground to store forty mobile suits and two Gundams. One can surmise that Quatre has been a generous host and has treated Trowa politely, as the Maganac members freely repair and reload Trowa's Gundam while the both of them are left at their leisure.
What we get to see is what appears to be Quatre's music room. There, he is relaxed and finding pleasure in playing the violin. Trowa meanwhile keeps to the background, still guarded, with his arms crossed. He does seem to be listening to the other boy's performance.
To the delight and many fans of the pairing and obviously to Quatre, Trowa finally moves on his own regard. He opens the case he has been standing next to. It holds various musical instruments. Trowa selects a flute and then proceeds to play, accompanying Quatre's violin. The song they play together is "Sparkling Harmony" and in that moment, perhaps they are in harmony with each other spiritually.
The thing is, the next scene we see of them, Trowa is walking away from the villa. It is indicated he was leaving without a word to his gracious host. Quatre catches sight of him, however, through a window and speaks amicably despite his guest's rude exit. He is understanding and only then asks for his name.
Trowa remains his impassive and distant self, despite the previous time they were shown he had seemed to finally let down his defenses. He does, however, give Quatre the desired response. Quatre uses the name, attaching friend to it, and expresses his belief and desire that they will meet again. Trowa shows no change to this, and it is later on implied that he had not shared the sentiment.
When they end up at the same harbor when preparing to set off over the Atlantic, Quatre spots Trowa and follows him to a parking garage. Quatre greets him kindly. Trowa, however, purses his lips slightly and eyes Quatre as though he is an annoyance. Undeterred, Quatre makes a remark on their unlikely meeting. Trowa can see where this is going and replies that he works alone.
This is false, however. As shown with Wu Fei and Heero, Trowa only will work with those he considers proficient soldiers. Perhaps the main reason he is unwilling to work with Quatre is due to his open nature and familiarity. He likely deems the other as acting unprofessional and thus could be a liability no matter how charismatic a leader and well-trained at fighting as he is. He will not take a gamble with such a wild card of a pilot.
Once again, Quatre is fortunately not put off by Trowa's cold reply and simply reflects his defiant posture back towards him and replies he works alone as well, but once more tries to persuade combining their efforts. Trowa again dismisses him and ends the conversation by abruptly leaving. Nevertheless, Quatre's methods prove to be ideal in the long run.
They do show up at the New Edwards base together, but that really says nothing in regards to whether their relationship changed at all from where they were last seen together. Still, neither get in each others way and that is all that seems to be required to satisfy Trowa in the given situation. In the end of the mission, all five pilots scatter with Wu Fei staying with Trowa only briefly.
The next time Trowa knows anything regarding Quatre's whereabouts is when he and Duo make a show of fighting against OZ and heading to space. Like Heero and Wu Fei, he understands and responds positively to the message. They all decide to return to space.
In the end, this all leads to the third of the most infamous events in the series: Wing Zero's debut. During this, Trowa has strangely taken a complete one-eighty when it comes to how he reacts to the thought and presence of Quatre. He acts as though he knows him well and trusts him implicitly. Where did all of this goodwill suddenly come from?
While many fan-fictions seem to like to look at their unplanned meeting at the harbor of Lisbon to be a fine area for them to have gotten to know each other better off screen, I think this change in Trowa was much slower. I doubt he had established this view of Quatre even after the disaster at New Edwards.
If anything, it was likely his time at the circus and traveling with Heero did the change slowly happen. His mindset was completely set to that of a soldier everytime he encountered Quatre before then. The time they were apart he was able to slowly learn from Heero and Catherine to look at others through the eyes of a person. Thus, somewhere along his time traveling with Heero across Europe to the time he crosses paths with Quatre again, Trowa finally recognized Quatre's actions for what they truly were: The moment they met Quatre saw him as an actual friend. Not as a possible underling, not a soldier, not an ally, not as a comrade... but simply as a human he wanted to be friends with.
Now, many can and have written Trowa off as dense and as stubborn as a mule considering how long it took him to realize this, but one should once more take a good look at his upbringing. His entire life he had no name to truly call his own. Even as of taking up "Trowa Barton" as his name, it is only an alias. Always "No-name", he grew up in an environment that led him to believe that he was only useful as a tool. His only worth rested in his efficiency as a soldier. Thus he always went above and beyond when it came to being a soldier to ensure he would not be discarded as well as to survive. Having lived like that, how was he supposed to recognize Quatre's actual goodwill and sincerity to be true?
The Mercenary Corp. that raised him may have looked upon him fondly, but they unfortunately didn't teach him how to make friends and recognize them by the looks of it. They were to see him as a fellow soldier, not a little brother or child. The fact that he was such a brilliant soldier likely made it even harder for them to even recognize him as a child. Thus, even if any of them would have been happy and willing to teach him life skills of a normal civilian, the thought likely never occurred to them because No-name was always so quiet, serious and capable.
Yes, as far as Trowa is concerned Quatre is the first person to have ever looked at him not as a soldier and not as inhuman. Upon their meeting Quatre had seen him as a human worthy of a name, unlike his Captain. Quatre saw him as a person worthy of sincere friendship that asked for nothing in return but the same, unlike countless others.
Quatre Raberba Winner's existence established to Trowa that he was human. He was like Quatre: a boy trapped in an existence of constant war, but still needed to be able to stand on equal ground and connect with others. He could be a soldier and still have those things. He could reach out, trust and depend on others.