There’s no need to put ‘less agreeable’ in quotations.
The quotation marks there are to indicate that "less agreeable" is a quote from the language used in the article to define this type of personality.
The topmost sentence above is punctuated incorrectly, however; when a quotation is within another quotation, the single quote mark is used for the innermost quotation:
"get bothered by grammatical errors online have 'less agreeable' personalities"