Yrurtj

I'm playing a Firbolg character. The Firbolg's racial Firbolg Magic trait (VGtM, p. 107) allows them to cast disguise self once per short rest. As part of the character's backstory I want someone to be able to see through his racial ability to disguise himself, and instead be able to see his normal form so that rumours start spreading about this creature that's new to town.

I haven't been able to find any item that can make this possible, I'm new to the game so there's no surprises there. So before I scrap the idea and just go with "he ran out of time on the spell" or using dispel magic.

Is there a magic item, spell, etc., that can make it so that you can see through (specifically what's underneath) a disguise self spell?

There are several methods for defeating illusions such as Disguise Self.

Any creature with access to Truesight automatically detects illusions as false, so the True Seeing spell does just that.

True Seeing: This spell gives the willing creature you touch the ability to see things as they actually are. For the duration, the creature has Truesight, notices secret doors hidden by magic, and can see into the Ethereal Plane, all out to a range of 120 feet.

Another similar method is to acquire the Witch Sight Eldritch Invocation for Warlocks:

Witch Sight: You can see the true form of any shapechanger or creature concealed by illusion or transmutation magic while the creature is within 30 feet of you and within line of sight.

You could also use the Polymorph or True Polymorph spells to become a creature with Truesight, such as the Couatl.

Finally, you could find or create a magic item that grants Truesight, such as the Gem of Seeing, which also grants Truesight.

Gem of Seeing: This gem has 3 Charges. As an action, you can speak the gem's Command Word and expend 1 charge. For the next 10 minutes, you have Truesight out to 120 feet when you peer through the gem.

I think that your best bet is actually described in the spell descpription:

The illusion is only visual, so any sort of physical contact will only
interact with the real size and shape of you. Someone may also use an
action to make an investigation check against your spell save DC.

This could easily have any random person that makes physical contact with you know that this is an illusion.

Also, the duration is only an hour, so maybe you got to your inn a second too late and someone saw you.

Additionally, an anti magic field, true sight, dispel magic and a few others should also work fine, although much less common.

Eldritch Invocation

You have this invocation for 15th Level warlocks.

Witch Sight: You can see the true form of any shapechanger or creature
concealed by illusion or transmutation magic while the creature is
within 30 feet of you and within line of sight.

Truesight is an ability some creatures have that allow them to see through magic

Truesight: A creature with truesight can, out to a specific range, see
in normal and magical darkness, see invisible creatures and objects,
automatically detect visual illusions and succeed on saving throws
against them, and perceives the original form of a shapechanger or a
creature that is transformed by magic. Furthermore, the creature can
see into the Ethereal Plane.

Based on the description of what you want, I don't think you're going to find it. While there are many ways to see through an illusion, I can't think of any that the user wouldn't be aware of using. If someone casts true seeing and sees your true form, they're going to know exactly what's happening: the spell let them see through your magical disguise. There's no confusion or uncertainty: if no one else can see your true form, the one person who can see it knows exactly why.

However, there is a spell that might allow someone to become suspicious without giving up the whole game: detect magic. Suppose there's a low-level wizard who runs a magic shop in town. She regularly casts detect magic to determine whether items her customers wish to sell are magical or mundane. If you happen to walk by her open front door as she's inspecting an item, she might notice your aura of illusion magic out of the corner of her eye. However, that's all the information she'll get: that there is illusion magic on you. She won't know exactly what spell you're using. That seems like the perfect basis for a rumor: "something about this new guy isn't what it seems, but I don't know what."