Before I start answering the question, I'd like to provide a quick recap on quotation marks in general – or at least point at some articles we've covered here at fandom_grammar, as we've mentioned them on three separate occasions.
Back in 2007, the question of when it's appropriate to use single quotation marks was addressed by kay_brooke. In 2009, green_grrl covered quotation marks in the third part of the Grammar 101 series. And just a couple of months ago, theemdash posted a cartoon on "scare quotes". You can find each of these posts very easily by selecting the shiny new "punctuation:quotation marks" tag on the community, or you can just click on this link.
Now that you know more about the correct usage for quotation marks, how should you use capitalization with quotations? The Chicago Manual of Style provides guidelines for general as well as more specialized works, but I'm going to focus on general works. Fan fiction doesn't usually approach the realm of legal writing and textual commentary, so while guidelines for that sort of work do exist, they're not relevant to what Luke and Han may be yelling at each other while ducking incoming blaster bolts. If you do want to use that sort of quotation, I recommend looking at sections 11.19 and 11.63 of the manual.
Normally, a quotation should begin with a lower case letter when it is used as part of the sentence, even if what is being quoted is a complete sentence or something else which begins with a capital. For example:
If there was one thing Luke remembered from his time with Yoda, it was that he should "do or do not."If the quotation is more remote from the rest of the sentence, then the initial letter should remain capitalized:
Leia said that she was "sick of hearing about" Han's past successes at gambling.
Leia stared at Luke. "What do you mean Yoda said 'there is another'?"
Luke always remembered what Yoda had said: "Do or do not, there is no try."However, when the quotation being used forms a complete sentence when quoted, even if it's only a partial sentence in the original, then a lower case first letter may be changed to a capital.
Luke's world had shattered with the revelation that Vader was his father, but despite Luke's shouted denials, the Sith Lord was right when he'd said "You know it to be true."When what is being quoted is divided by an interrupting expression, the second part of the quote should begin with a lower case letter.
Admiral Firmus Piett winced as another asteroid slammed into the Executor. "Asteroids," Vader had said, "do not concern me, Admiral. I want that ship, not excuses." Piett rolled his eyes -- Vader wasn't the one who had to repair the damage. "Intensify forward fire power," he called to the gunnery officer seated in the command pit. "I don't want any more of those rocks hitting us."If you fall into the unlikely category of using block quotation in fanfic, then the initial letter of the quotation may be lower cased if the syntax demands it. This would normally only be if the quote is proceeded with something like "Han said that". It's far more common to capitalize when using block quotation.
In summary, there is no need to capitalize within quotation marks if the quotation is used as a part of the sentence. If the quotation is separate, or if the quotation forms a complete sentence as quoted, then the first letter should be capitalized.