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.
