I have a question about quoting.

If I have a sentence like:
Something something something something "something". Do I quote it this way?

"Something something something something "something"".
Actually what confuses me is the 2 apostrophes at the end.Is that possible?

No, if you have a quoted word or phrase within a quotation, you use single quotation marks, which can also be apostrophes. Here are a couple of examples:

He asked, "When will you be here?"

She said, "My brother asked, 'When will you be here?'"

http://grammar.ccc.commnet.edu/grammar/marks/quotation.htm
Scroll down to Single Quotation Marks for more examples.