While there is peace...let there be romantic mishaps!
When the night sky wears the moon as its pendant, I shall await you at the dock.
Eiko, the little girl from Madain Sari, has a bit of a crush on Zidane. She figures that the key to his heart is to write him the most romantic letter ever! Enlisting the help of the sufficiently bookish enough looking Doctor Tot, the above letter is penned and she sets off to get her man.
Unfortunately...she gets a little stuck. Thankfully, Baku (the leader of Tantalus) is wandering around the castle (for some reason?). Eiko asks him to deliver the letter to Zidane, which he agrees to do. However he gets into a little argument with Steiner and in his agitation, drops the letter on the steps at the dock in the courtyard. Steiner and Baku go on their separate ways, neither one noticing the letter on the ground.
Beatrix is morose, despite the fact that Garnet is safe and the kingdom is no longer at war.
Beatrix: "Alexandria is at peace again... Yet my heart is full of sorrow. Why...?"
There are, of course, many reasons why she would be full of sorrow – a topic that will be covered more later. She doesn't have time to dwell on her sadness, however, because she notices the letter on the steps and assumes it was something dropped by Steiner. After all, he had just been there. So she is surprised to read it and discover that it's a love letter - and now she wonders if Steiner left it there for her to find.
Eiko is waiting at the dock and regretting sending the letter - she sees that Zidane has feelings for Garnet and knows she has no chance. When she hears some people approaching she hides - but it turns out to be Blank and Marcus of Tantalus. Blank now holds the letter - it fell from the floor above him, and he believes some shy lady wants to meet him. However, when they hear someone else approaching, they both have a little panic and also hide. As they flee, Blank drops the letter, because apparently Alexandria is full of people with butter fingers.
This time, it's Steiner who has arrived. He's on his patrol and thinking about castle security when he comes upon the letter. He's adorably flustered.
Steiner: "M-M-My goodness!
Th-This is a love letter!
Who wrote it to whom!?
Did someone drop it here, knowing that I would pass by?
Who could it have been...?"
And once again, enter Beatrix.
When he sees her, Steiner seems to assume that Beatrix was the one to send the love letter. They step closer to each other - completely oblivious to their audience who are loving this.
Steiner and Beatrix step closer still, and it very much looks like there's going to be a kiss...
...until Baku announces his arrival with am extremely loud sneeze. Beatrix and Steiner turn away from each other, and the moment is lost.
But perhaps the interruption was a good thing. It gives the two of them time to consider their changing relationship, especially considering the slightly antagonistic vibe of it before. While it's certainly possibly they held hidden feelings for each other before this, it doesn't really seem apparent. What I think this event with the lover letter does is open them both up to the possibility, and marks a change in their relationship.
Before all this business with love letters, eidolons, and crazy queens, Beatrix and Steiner were rivals. As previously mentioned, Steiner did best Beatrix in a practice fight - possibly by sheer luck. Beatrix is a proud person, and could be holding onto that loss with a little grudge.
However, in the grand scheme of things, Beatrix and her squad are the ones who have far more prestige and respect. Beatrix's soldiers make several comments here and there about Steiner and his Knights of Pluto. Right near the start of the game, one of them states, "I see the Knights of Pluto are as hopeless as ever." After the assault on Cleyra, another one says the following to Beatrix:
"Your supremacy is without question now. Steiner and his Knights of Pluto are nothing compared to you!"
While Beatrix quickly shuts the soldier down, it does lend truth to Steiner's earlier thoughts about Beatrix always trying to one-up him. Despite her superior position, Beatrix does still see Steiner as a rival.
On his part, Steiner is definitely concerned about how he is seen by Beatrix and her squad. There's probably also jealousy there, as there's no doubt that Steiner is as dedicated to his work as Beatrix and doesn't seem to receive the same recognition. It's not helped by the snide comments made by her soldiers. At the start of the game while he is searching for Garnet in Alexandria castle, his thoughts turn towards Beatrix.
Steiner: "...we will be the laughing stock of Beatrix and her retinue!"
It must be incredibly frustrating for him. And it makes it interesting that after Garnet has had her eidolons extracted from her and she, Zidane and Vivi are attempting to escape Alexandria, Steiner decides to stay behind. He decides to stay behind and assist his biggest rival - one who just easily beat him and three others in battle without seeming to care that he was someone she's worked with for years. He's also willing to leave Garnet for this. And leave her with Zidane, too, of all people. Steiner has made no secret of the fact that he is unimpressed with Zidane.
But stay he does, and he joins Freya and Beatrix in battle. There's an argument to be made that Steiner does this because of hidden romantic feelings. But I personally prefer to think it's a sign that he's getting over his jealously of Beatrix and his feelings of inferiority, along with the internal conflict he was feeling about his duty. So by the time Brahne dies and Garnet returns to Alexandria, both he and Beatrix have settled a lot of their antagonistic feelings. So no, maybe not love yet, but shall we see what happens next?