Period-quotation mark-comma is not correct. What is correct depends on what style you're using.
Strunk would suggest that, if it is a formal quotation being used as evidence, you need to re-write the sentence.
The APA Style Guide has a rundown of British versus American usage here.
This article from Oxford Dictionaries (which actually focuses on single versus double quotation marks) notes that "Any punctuation associated with the word or phrase in question should come before the closing quotation mark or marks".
I would say that, assuming the period is in the original quotation, you'll want to do it this way:
His quote, “The most dangerous thing is illusion”, points to the idea
that it is not only the targeted individual that is harmed, rather the
illusionist and often, bystanders who are unaware of the situation
they are in, are harmed as well.
Putting the comma outside the quotation mark indicates that it's your comma, not the original source's comma. A real stickler for accuracy might suggest that you need to add an ellipsis:
His quote, “The most dangerous thing is illusion...”, points to the idea
that it is not only the targeted individual that is harmed, rather the
illusionist and often, bystanders who are unaware of the situation
they are in, are harmed as well.
However, I think that's clunky and suggests the sentence goes on, which it doesn't. The APA Style blog also deems that incorrect.
Best Answer
The quote is not ungrammatical. It does, however, some some archaic constructions which are no longer heard in modern English.
The main stickler is this bit:
The phrasing "needs not" is no longer used in contemporary English, but in a literary register it is an acceptable variant for "does not need".
An additional wrinkle is that many speakers prefer to drop the -s when using this construction, treating the verb need as a modal auxiliary. This would give:
The latter is somewhat more common, but both constructions are well-attested. For a bonus, here is the N-gram chart comparing "need not" with "needs not" since 1830:
http://books.google.com/ngrams/graph?content=needs+not+be%2C+need+not+be&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=
As the chart shows, in the early 1800's both variants are reasonably common, though even then the variant with needs is much less popular. As time goes on the needs variant becomes increasingly less common, because do-support for negation becomes more and more exceptionless. The need variant survives somewhat better due to its analysis as a modal auxiliary.