Is there any lore justification for being able to do previous expansion's sidequests?

[Cleretic]

First of all: the direct answer is no. We can play in the space of making up a justification, but we must first admit that there isn't one on an official, written-down level. That's just how the game is, and you just have to accept it. 90% of Ala Mhigo will always be frozen at the assumption that Stormblood is either still going on or just wrapped up.

Now, as for making up a solution, that's gonna depend on why you want this, and in many cases, also on what quest we're talking about. You're on Balmung, so I'm gonna assume this is for roleplaying purposes, and as a roleplayer myself I think I can speak to a few ways to look at it.

• First of all, in a roleplaying context it's actually a little rude and overbearing to declare in open roleplaying that you were involved in a canon event in the first place; this is sort of an extension of the general community etiquette of 'don't RP as the Warrior of Light', because there's only one of that guy and we can't all be him. This is very 'inside baseball', though.
• Consider that, perhaps, you didn't do it: depending on what you need of that scenario, maybe your character played a different role, or studied it afterwards, or were around for a coincidentally similar thing. It lets you incorporate the story without explicitly taking credit for it, and can give you a bit more control over your character's relationship with it. Again, this is very dependent on what you need.
• Alternatively: what if you ALREADY did it? Screw linear time; if it makes sense for your character to do a questline, then they did it back when it was roughly taking place. You want to say your character was studying to be a monk back before the liberation of Ala Mhigo, like in the ARR story? They already did, you can just say that! You're the character's writer, what you say goes.
• And on an entirely different level of 'screw it': who says the events of the quest actually did happen when the game sets them? Sure, some amount of sidequests reasonably have to happen at that time--Eden can only happen immediately after Shadowbringers, for example--but smaller ones especially can be very easily shifted, kinda by design to permit people to play whatever part of the game they want, whenever they want. Who says Machinist has to take place during/near Heavensward? It could've happened last week.
• If you really want to dig, technically the Echo was a weird time jump like what you first described back in 1.0, but... well, there's a reason it's not written like that anymore: it was really frustrating and very confusing. Nobody's gonna stop you from just using that, but I suspect the main thing that'll happen is that you'll learn why they don't do that anymore.

Those are all very general suggestions, though; more specifics on why you want this, and what quest you're looking at, can probably help shape out an angle better suited for your needs.