Is an article (a, the) considered a part of the noun, when it comes to text formatting?

Let's say I want to bold the noun "emotion" in the text. Which is better way to do it?

1. An emotion is the feeling in the heart.

2. An emotion is the feeling in the heart.

So, should I consider articles (a, the) as parts of the nouns when it comes to text formatting?

It depends entirely on if the article plays a semantic role in what you're emphasizing. Sometimes, you don't even emphasize the noun at all:

You aren't the Keanu Reeves are you?
You can't eat only one apple can you?

If you mean to emphasize the noun, there is nothing saying that you have to also emphasize the article too. Only do so if it makes sense to do so. It also depends on why you are emphasizing something. Are you putting stress on its pronunciation—or are you using the emphasis to define the word as a word?

The word emotion is . . .

But, perhaps:

It's better to have an emotion than it is to have no emotion.

In short, emphasis signals your intent. (Note that I'm using italics rather than bold text, but the result is the same. Any number of stylistic devices could be used to emphasize the text.)

Barring any other context, my personal opinion is that the second sentence in the question looks better—at least it makes makes more sense to me that it's the noun, and only the noun, that's playing a particular role. However, if the sentence immediately followed another one—perhaps one that made the same kind of distinction as the last example of my own above, then, in that context, emphasizing both the article and noun could make sense.

So, don't consider the article to be part of the word. But do be aware of exactly which words make sense to emphasize—there's no reason why you can't emphasize more than one if that's the best way of representing what you're trying to express. However, it's certainly not a stylistic necessity.