Talk it out with your players
This is the most routine piece of advice on this stack, but for a good reason. It's especially relevant in this case because there hasn't been an undesirable in-game consequence yet-- it's all about your concerns regarding what might happen in the future.
So talking to your players about your concern, why it's on your mind, and why it would be bad for the game for things to end up that way is the best way to get that information to the players early enough to avoid problems. Further, because the situation has apparently caused tension between the characters (and/or players) it will be valuable to get information on how upset the characters (and/or players) may be about what happened. That way you can make sure that potential solutions address what has actually upset people.
It's also worth noting that this is an out-of-character discussion. It's fundamentally meta to the game because, however in-character people are at the table, what you're talking about is the characters behaving differently than they "naturally" in order to serve the broader game and its needs.
Consider not letting players do things like this freely
It's a fairly common ruling, though by no means universal, that PCs have limited ability to dictate the thoughts, feelings, and cooperation of other PCs in this way. If we presume that the Tiefling deliberately chose not to share their backstory, being coerced by the wizard to do so may have really cut into how the Tiefling's player wanted their story to unfold. And if something similar might happen again, it could feel like the Tiefling is adventuring with an enemy wizard they have to defend against at all times.
As DM you have unique knowledge of and responsibility for game mechanics, setting, and plot. You can use these to manage many situations like this one. In this specific case, how reasonable the wizard's suggestion was could limit how much information the Tiefling could be prodded into revealing. As DM, you get to decide if the wizard's suggestion is reasonable enough to work as they hoped. You can also speak with the Tiefling's player to work out how they would interpret the suggestion-- if revealing backstory element A is tantamount to suicide, in the Tiefling's eyes, then they can't be Suggestion-ed into blabbing about it.
You, as DM, also get to adjudicate effects. If the wizard had cast Suggestion on an NPC, you would determine everything about the NPC's response in light of the spell. In the scenario that happened at your table, after the wizard's spell was determined to be successful you could pull the Tiefling's player aside to help them determine what information the wizard would get. Some people aren't good at figuring that sort of thing out on the spot, and your supervision could keep the spell and the response proportionate and appropriate. And you can certify the truth of what the Tiefling says to the wizard, within the limits of the spell and the character's stats. Players get to choose what their characters think and do. They don't get to choose what happens.
Everything the players do is plot
Even if it wasn't a part of your main campaign, if this sort of inter-character tension arises you can weave it into the story or create a new subplot. As written it sounds like the wizard wanted something and compelled the Tiefling against their wishes. That's an annoyance with no attachment to the story, so it becomes an irritating reminder that the wizard got their way in response to a whim. No stakes, no significance, just a conflict between two characters that one character won completely, and the other lost completely.
It doesn't have to be that way. Maybe it turns out that the Tiefling wasn't just being taciturn, but that the information the wizard forced them to reveal is in some way dangerous for the wizard to know. Maybe it's illegal to magically coerce people in that way in polity they're all in, and so the wizard gets fined (or worse) for their gall. Regardless of the specific circumstances, you can design a story element that gives the wizard an obstacle or disadvantage they wouldn't have otherwise had, and ideally that will ultimately drive the wizard and Tiefling to work together.
I've taken this approach (with other game systems), and it can work well. The plot simply demands that players not dwell on the event, or nurture a grievance over it, and also provides a bit of disincentive for players to ignore the preferences of other players.
Get ahead of likely problems, and recognize meta-issues for what they are
It can be tricky to spot when they're coming up, but it's not unusual for a group of strangers freely coming together to risk mortal danger to want to know a bit about one another. So you can trust your fellow party members, or at least have an idea of how much you can trust them. I, personally, would probably not plan a dungeon crawl with a Tiefling on the assumption that the Tiefling is probably cool and that our interests probably align.
Many gaming groups handwave this away, implicitly accepting that the characters will journey together because their players have chosen to play a game together. Others don't want to gloss over the rationale for the party forming and continuing to exist. You can discuss with players reasons that they are adventuring, or why they might want to team up with the other players, before sitting down at the table. You can assign or organize plot elements for individual characters such that they have little choice but to work together. You can tell your players that you're not interested in forcing their party to remain cohesive, and that they need to figure out a way for their own characters to justify staying with the group.
In any case, you can tell your players that you're concerned that intra-party tension is undermining both the fabric and the fun of the game because if they won't play together, the game itself can become difficult or impossible to run, not to mention less fun for all involved. Since the fundamental expectation for a party of PCs in a game people are playing together is that they will be a party together, you can insist that your players find a way to make that happen for their characters. That may involve one or more players rolling a new character.
You can also force it via plot device, if necessary (if a deity curses them such that the characters die if they're more than a mile apart from each other, they'll be adventuring together no matter what their relationship). In session 0 of any new game I run, I explain to my players that if they can't keep their party together I will intervene to make it happen once. They might not like my solution, but they'll be stuck with it at that point. If the players can't figure out a way to play together, then the group won't be able to play together, which is equivalent to the game simply dissolving.
