I am quite certain, without checking, what should be done in British academic style, in which I was trained, but I had to check to make sure that it was the same in American style, which, of course, is sometimes different in matters of punctuation. On this particular point, it is apparently the same.

1. If the exclamation mark belongs to the quoted matter, as it does here, it goes inside the closing quotation mark, as you have it.

2. If this coincides with the end of the sentence, no point is added after the closing quotation mark in American style, though in British style a point is added if the quotation does not constitute a complete sentence:
American: I heard him say "I can't stand it any longer!"
British: I heard him say "I can't stand it any longer!".

3. However, in the case you are referring to, there is a parenthetical reference after the end of the quotation, and you want to know whether to add a point after that reference. The answer is that you do need to if the quotation is run-on in the text, but with a block quotation you do not. This is so regardless of whether the quotation itself constitutes a complete sentence (as in your example) or is merely part of a sentence, and this is true in American as well as British style. The reason is that the parenthetical in-text reference belongs to the same sentence as the quotation; it is not a separate sentence on its own. Therefore the end of the quotation is not the end of the sentence, and the closing punctuation at the end of the quotation (here an exclamation mark) cannot serve as the closing punctuation of the sentence. So your second alternative is the correct one:

“Since I have vowed to fast for the Merciful, I cannot speak to any mortal today!” (Qur’an 19:26).

However, if it were a block quotation (used for longer quoted passages), there is no punctuation after the parenthetical reference at the end (and the quoted matter, set off and usually indented, is not enclosed in quotation marks):

"We find the following statement in the Qur'an:

<blockquote>Since I have vowed to fast for the Merciful, I cannot speak to any mortal today! (Qur’an 19:26)

[Text continues]"

I hope this makes sense!

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Note added at 2 hrs (2014-05-02 08:38:08 GMT)
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It's not easy to find an explicit answer to this question in online style guides. There is no doubt at all that punctuation follows in in-text citation when the quotation is not a full sentence, but the precise case you raise, in which an in-text citation follows a quotation that is a full sentence, does not seem to be addressed, perhaps because making quotations full free-standing sentences is not generally recommended anyway. The following, from the Purdue OWL guide, is about MLA style, but I am confident that Chicago will be the same on this:

"Punctuation marks such as periods, commas, and semicolons should appear after the parenthetical citation. Question marks and exclamation points should appear within the quotation marks if they are a part of the quoted passage but after the parenthetical citation if they are a part of your text."
https://owl.english.purdue.edu/owl/resource/747/03/

The statement is unqualified; it doesn't say that punctuation should follow the citation unless the quotation is a full sentence. Lower down the same page they present a block quote example with no point after the reference, which bears out what I've said.

There's also a relevant Chicago Q&A here (see the eighth question, beginning "With the author-date system, I know that punctuation is supposed to follow the reference"):
http://www.chicagomanualofstyle.org/qanda/data/faq/topics/Pu...