I'm a D&D fan and Magic: The Gathering fan.

I'd heard that Wizards of the Coast says that Magic: The Gathering planes are Dungeons and Dragons planes. I was so excited and full of idea of adventures.

In my research, I didn't see anything about planeswalkers from MTG.

Is there lore and/or written rule(s) supporting MTG planeswalkers? If so, what/where is it?

Yes, there are.

The rules for PC Planeswalkers are found in the appendix of the Plane Shift: Amonkhet document, available for free download on the WotC website. The rules themselves can be found in two paragraphs on the final page of the document, with the rest of the appendix discussing the implications of playing a Planeswalker PC.

Basically, a Planeswalker PC can planeshift to another plane by using a minute of concentration akin to a ritual spell casting, arriving at a location on the plane determined by the GM. Additionally, if reduced to 0 HP, they can make a Charisma save with a DC equal to the amount of damage they've taken to planeshift away rather than take the damage, with the GM determining what plane they arrive on.

The Planeswalkers are mentioned in Guildmaster's Guide to Ravnica

The History of Ravnica section of the book contains the following passage:

Jace is a Planeswalker, with the ability to travel from world to world, and his attention never remains focused on Ravnica for long. Thanks to his involvement with other Planeswalkers, he spends extended periods of time away from Ravnica.

However.... there are no official rules for how Planeswalking works in D&D 5th edition.

But you can find rules for how to use PC Planeswalkers in the Appendix of "Plane Shift: Amonkhet" (as explained in nick012000's answer). Keep in mind though that the Plane Shift Series is self-admittedly (as found on the bottom of the Introduction page):

not fully tempered by playtests and design iterations.

As such, using these published but unofficial rules should be a discussion with your GM and the rest of the players (see this question for what counts as official rules).

The worlds of Magic: the Gathering are not "planes" in the same sense as planes are described in D&D 5e. Instead, they are more similar to the different worlds that D&D campaigns can exist within. Some example of these include the Forgotten Realms, Greyhawk, Eberron, and Dark Sun.

D&D canon

Planeswalking in official canon is likely a form of Spelljamming. In the campaign guide, Spelljammer, all of the worlds of D&D (and their respective Material Planes) exist within crystal spheres that lie within a medium called the phlogiston (not dissimilar from the Blind Eternities in MtG lore). Spelljamming typically requires advanced arcanomechanical vessels called spelljamming ships.

The levels of magic involved in these vessels is far beyond the reach of traditional adventurers until the upper reaches of the Character Advancement table. As such if Planeswalking were ever represented as an official character option, it would likely be an Epic Boon of sorts.

Note: There are entities from the Spelljammer setting in official D&D rulebooks such as the Giff found in Mordenkainen's Tome of Foes.

There are no specific rules for Planeswalker characters in D&D, probably because they aren't really needed

If we reduce the MTG Planeswalkers to their key properties, they are really powerful entities/spellcasters and have the specific (and unique, in the MtG muliverse) ability to travel between planes (there's some difference in the cosmologies, so it's maybe more like traveling between worlds in D&D, but we can ignore that).

D&D already has wonderful ways to represent such characters: high-level spellcasters with access to plane shift (and/or gate. That spell is available to most spellcasting classes (Cleric, Druid, Sorcerer, Warlock, Wizard), and so having that ability be part of the class ability makes it fit neatly into the D&D rules without wonky extra rules or options.

The Plane Shift: Amonkhet article specifically recommends doing this in its Planeswalkers and the Multiverse appendix, i.e. playing high level characters and using the plane shift spell to emulate planeswalking. However, it does not provide any specific rules for doing so. The closest is giving each character the ability to cast plane shift (self-only) under some (vague) constriction. It also advises that you talk with your players about them following up on plot hooks and not just going wandering about the multiverse. (Thanks nick012000 for pointing this out.)

If you wanted a D&D feature/option which could represent the Planeswalker 'Spark' (a MtG term), it would need to be independent of classes and races so it could still represent the variety of Planeswalkers possible, and the ability to cast a 7th-level spell, even if restricted to self-only or similar, is probably beyond the 'power-scope' of feats and backgrounds, and over in the territory of Epic Boons (see DMG p. 231) instead. In fact there is a specific Epic Boon which does pretty much that:

Boon of Planar Travel

When you gain this boon, choose a plane of existence other than the Material Plane. You can now use an action to cast the plane shift spell (no spell slot or components required), targeting yourself only, and travel to the chosen plane, or from that plane back to the Material Plane. Once you use this boon, you can’t use it again until you finish a short rest.

While these Boons are intended as progression/advancement after level 20, there is nothing to stop you from granting them earlier, nor from simply giving them at character generation if you would like them to already be planeswalkers.

As for igniting the Spark, PS:A does suggest an optional rule that:

a Planeswalker who is about to drop to 0 hit points can make a Charisma saving throw with a DC equal to the damage taken. On a successful save, the character instead takes no damage and planeswalks away. It’s up to the DM what plane the character ends up on, because this isn’t usually an intentional process.

You can probably extend this rule to other traumatic events, with an appropriate DC (if desired). As a general note, be careful with your table's boundaries if you're planning to expose them to trauma.

As for lore, any lore about MtG Planeswalkers are going to be found in MtG-specific material, not the 'core' D&D stuff.