I love the blue that it’'s called “Vermeer blue”.

Since you're referring to a specific type of blue, you can use "the" here (this points out that you're talking about one type of blue, rather than blue in general)

Vermeer blue is made from Lapis Lazuli, thea pigment.

"the" is technically ok to say here, but only if you can assume your audience knows about lapis lazuli and you're just reminding them of what it is. In any other case it would feel more natural to say "a pigment" to introduce it.

AtIn that timee past, thate pigment was as expensive as gold and very valuable.

At what time? It feels unnatural to go from the previous sentence that make no mention of time to this sentence that suddenly expects a reader to know about a specific time. Instead consider a phrase like "in the past".

When I searchgoogle about Vermeer blue, it says, an art exhibition featuring the painting "Girl with a Pearl

"When I search for "Vermeer blue"" would also work here, but the generally accepted verb for searching the internet is "to google" these days. Even when people don't use Google, they still say they're "googling" something.
"the painting" is technically not needed but adds some extra clarity for people who might not already know it's a painting.

Earring of" by Johannes Vermeer art exhibition will be held in Osaka from the end of August to the end of September 2026.

When someone creates something we usually say it's "by" them. Eg: Great Expectations, by Charles Dickens

What iIs yYour fFavorite cColor?

Proper capitalization in your title.

I love blue that it’which is called “Vermeer blue”.

"which" is the correct relative pronoun here.

Vermeer blue is made from Lapis Lazuli, the pigment.

More natural phrasing without the extra article.

When I searched about Vermeer blue, it says,id that an art exhibition featuring Girl with a Pearl

"searched" Past tense matches the action.
"It said that" Past tense consistency.

Earring ofby Johannes Vermeer art exhibition will be held in Osaka from the end of August to the end of September 2026.

"by Johannes Vermeer." Correct preposition for authorship.

I wonder if that is an original?

"I wonder" already makes it indirect, so no question mark is needed.

I love the shade of blue that it’s called “Vermeer blue”.

Vermeer blue is made from Lapis Lazuli, the pigmenta pigment called Lapis Lazuli.

Just slightly more natural phrasing.

When I search about Vermeer blue, it says, an art exhibition featuring "Girl with a Pearl

Connecting with the next line

Earring of" by Johannes Vermeer art exhibition will be held in Osaka from the end of August to the end of September 2026.

Connecting to the previous line.

I love a blue that it’'s called “Vermeer bBlue”.

While colors are usually not capitalized, since this is a color with a proper name, "Vermeer Blue" is capitalized.

Vermeer bBlue is made from Lapis Lazuli, the pigment.

When I search aboutlooked up Vermeer bBlue, it says, anwhat came up was a Johannes Vermeer art exhibition featuring "Girl with a Pearl

This correction connects with the next one as one sentence.

Earring of Johannes Vermeer art exhibition" that will be held in Osaka from the end of August to the end of September 2026.