I currently have a set of characters in which two of them have had (and still do have) history together, however I do not intend to drop the ball on the reader and write down their backstory because it does not seem to fit the narrative. They're side-characters that aid the true protagonists and due to their relationship being a secret they're trying to keep it to themselves, but I'd still like to hint that relationship in some shape or form without it being too obvious.
I'd fall back once again on the Iceberg Theory. Go ahead and fully flesh out their entire relationship --for your own benefit. But only include the details in the story that would naturally come to the attention of the POV character.
I like the answers I've seen so far.
I may also suggest something subtle like one of them being thoughtful or cautious regarding how they answer a question about the other, such as their current location or activity, or even their favorite color or what would be a good gift. They could be concerned that showing too much knowledge of the other might reveal their secret.
Another, somewhat less subtle, option would be to feature one of them denying another character's suggestion that they might make a good couple. But it can be tricky to keep that one subtle without masking it as a moment of comedic relief.
If my experience with fan fiction has taught me anything, it's that readers are able to "ship" (as in, see a potential relationship between) even characters who never had a scene together and are unlikely to ever have met.
It helps if the characters are portrayed in a way that romantically inclined readers would consider them a good match. (For example, they could share a common interest. Or have similar personality, or complement each other's strengths and weaknesses in the way they interact in the workplace.)
You can also drop small clues hinting at them spending more time together than would be expected or talking about topics beyond whatever is necessary for their official relationship. They're probably giving gifts to each other, they're likely to show an interest in the other's well-being, etc.
Maybe character A likes to read and character B doesn't, yet it's B you meet in the bookstore. Maybe one character always seems slightly more knowledgeable about what's going on at the other's workplace than you'd expect. Maybe character A has a specific type of flowers growing in their garden and, at some later point, character B has a vase of the same kind of flowers. Maybe, at one point, both characters coincidentally have a cold at around the same time. Maybe there's a turn of speech (a quote, an expression) they both use to refer the same kind of thing. Some of these are probably already too heavy-handed.
Don't make a big deal out of any of those, but over time the clues will just add up. In fact, you might find yourself having to cut it down when you find that too many of your test readers pick up on the relationship too early, or place too much importance on it.
I don't see it mentioned, maybe hinted at in other answers (kind of adding to the iceberg theory here), but people in relationships often have shared experiences. More often than not, even friendships, will make references to something that happened to them a long time ago, an inside joke of a sort.
Innuendo, is an indirect intimation to a person or a thing (not just sex as it's normally used).
An example: one could say "It's like that biker at the concert," which means nothing to anyone else but would make the other laugh. So the shared experience gave you a way to show they know something the others don't and may have more going on without directly saying it.
An alternative is to change your definition of "secret". For example, I knew two professors that were sleeping together and exclusively (one was a friend of mine outside of work), but very few of our colleagues knew the extent of their relationship. The department chair had been told by them (as required to deal with conflicts of interest), but is prohibited from telling anyone about it.
At work they were friendly, but not intimate at all, not even touching. They appeared to be friends. Laughed at the same things. Went to lunch together but usually with a group. They did not mention dating or intercourse.
Now if somebody asked him what her favorite color was, or favorite band, or whatever, he would answer truthfully. If somebody asked him if they were dating or sleeping together, I'm pretty sure he would reply that romance and sex were unprofessional and inappropriate topics of conversation.
I heard a gay actor say a great line during an interview: "I live my life as an open book. That doesn't mean I have to read it to you."
For these two professors, to what extent is their relationship a "secret"? They don't lie about it, they won't deny it, nor will they confirm it. They won't pretend they don't know each other.
Another great line, this time from a sitcom: Two guys are sitting together. One of them, we (the viewers) know has committed a crime.
Guy A: "Did you do it? Just tell me."
Guy B: "Look. You're my friend. I'm not going to lie to you." (long pause, Guy A gets impatient.
Guy A: "So? Did you do it?"
Guy B: "So that's my answer! I'm not going to lie to you!"
Your "secret" can be not so much an intentional deception or lie, but just not revealing their personal feelings and history. And in that vein, you can portray them as best friends without crossing the line into things they would only do or say as lovers. For example, they can seek other out in their idle time and at meal times, be seen laughing together often, finish each other's sentences. They like to be together. Like best friends, they know each other, and how they think. They are concerned specifically for each other's safety. If one is sick or injured, the other cares for them. But they keep their private life private; not by lying about it, but by not volunteering any information or answers about it. From outward appearances, you might not be able to tell if they are siblings that love each other nonsexually, or a pair in love.