I want to know if there is a way my Fighter can make a magical artifact. Is there a certain thing listed in any book that says a player can find something to forge a magic artifact?
No. Artifacts are generally used by the Dungeon Master as plot devices.
The guidance for artifacts in the Dungeon Master's Guide states:
Characters don’t typically find artifacts in the normal course of adventuring. In fact, artifacts only appear when you want them to, for they are as much plot devices as magic items. Tracking down and recovering an artifact is often the main goal of an adventure. Characters must chase down rumors, undergo significant trials, and venture into dangerous, half-forgotten places to find the artifact they seek. Alternatively, a major villain might already have the artifact. Obtaining and destroying the artifact could be the only way to ensure that its power can’t be used for evil.
Characters typically don't even find artifacts, much less "make" them. Artifacts are generally used by the DM as part of the story, often as a maguffin of sorts. There are no player-facing rules for obtaining artifacts.
With the DM's permission a character could spend downtime creating a Legendary magic item, which, using the guidance from Xanathar's Guide should take 50 work weeks and 100,000 gp. Again, you definitely need to work with the DM to do this, as 50 weeks of down time is a lot of down time.
The answer is NO, but talk to your DM
If you want a specific item, you can't make it yourself. You could however with the blessing of your DM to go on a quest finding that item.
When I DM I always listen to my players. That means that if they feel they want an item and its balanced to their level (I won't give a level 1 fighter a sword of ancient dragon summoning) and the game (if its medieval settings you won't get a laser rifle).
If you can make sense with what you want the weapon to do, be specific how you want to use it and be passionate about why you need it, you might get your DM to allow you to somehow get it.
Pick that weapon instead of a feat when you get the next feat.
Or maybe you'll have a quest for that item or something similar.
Artificer already exists and that's about the most you can have with the game staying balanced.
If you want to focus on one item, as a DM I'd add big limitations to that:
Like make you only be able to use that weapon if its a weapon, or make you pay for the artifact with something that'd inconvenience you like suffer damage from contact with gold silver and platinum thus rendering you unable to use coin and you'd have to start either stealing, or use gems as payment which you can't do anywhere... I'd find something that'd challenge you just as much as you artifact help you. As long as the game is fun anything is possible but there's no point of making rules for that because it all comes down to you, your DM and your party. Therefore there are no rules for making artifacts as a player.
Not an artifact
‘Artifact’ is a very special term in D&D (not just 5e, but also in 4e, 3.5e, and possibly early versions though I have no reference for those). It specifically refers to a magic item that is functionally equivalent (in the context of the setting in which it exists) to a superweapon in terms of power. In most cases, artifacts are also unique, and most of them are ancient. Examples from other media that meet the normal criteria used in 5e within their respective universes are the Rings of Power from Tolkein’s Lord of the Rings, the Deathly Hallows from Rowlings Harry Potter, the Dragon Balls from Toriyama’s Dragon Ball, the twelve Zodiac Keys from Mashima’s Fairy Tail, the eponymous sword from the Soul Calibur games, the Master Sword from the Legend of Zelda games, the Infinity Stones from various Marvel sources, and most named weapons from various mythologies (such as Mjölnir, Imhullu, Excalibur, or Kusanagi-no-tsurugi).
Because of their overall power, artifacts are usually crucial plot points in adventures, and often are not actually interacted with by the characters, let alone owned or created.
Even ignoring the fact that rules as written there is no way for a player to create an artifact (at least, not one that they would be able to utilize the abilities of, if you make a big enough name for yourself maybe some magic item you pick up will eventually become an artifact more than a thousand years from now), no sane DM is likely to allow a loophole for this because the power level is so heinously unbalancing to the game.
Possibly a regular magic item.
Regular magic items are something rather different. They are still generally very difficult to get (5e is very much an inherently low-magic system compared to previous editions), and there are no standard rules for creating magic items as players unless you are playing an Artificer. Depending on your GM, they may allow some kind of house rule to let you make a magic item under other circumstances, but it will likely be no less expensive or time consuming than finding one randomly (and will probably actually be more expensive and time consuming, because you’re trying to get a specific item).
However, that’s something you would need to discus with your DM, and probably hope. Crafting in 5e is already not used much in many campaigns (due to being a downtime activity, and a lot of campaigns not really involving much downtime), and this would be something that goes well beyond ‘normal’ crafting.