Is it permissible to quote from the Holy Quran, such as His —the Almighty’s— saying: "Indeed, it is from Solomon, and indeed, it reads: 'In the Name of Allah, the Most Gracious, the Most Merciful'" in an inappropriate context or occasion, merely due to a similarity in names?

Scholars have differed regarding the quotation from the texts of revelation. The majority of scholars permitted it if it was for a good and lawful purpose, but forbade it if it was for a corrupt purpose. Al-Suyuti divided quotation into acceptable (in sermons and admonitions), permissible (in speech, letters, and stories), and rejected (such as embedding a verse in a jocular context or attributing it to other than Allah). The example mentioned in the question is incorrect in its meaning, as it does not apply to the intended purpose. It places the quoted text out of context, and thus falls under the rejected category. Therefore, if the quotation is for a praiseworthy purpose and consistent with the meaning, it is good; otherwise, it is disliked or forbidden.