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I understand you should always follow a quotation with a citation in academic writing, but how close together can two of the same citations be? Here is the sentence in question:

“BitTorrent traffic accounts for 18% of all internet traffic”, [3] two thirds of which is estimated to be illegally distributed copyrighted material. [3]

Should I even include the second citation? It also comes from the cited source, but having two identical citations so close looks clumsy.

So why is the first half of that sentence a quote and the second half your paraphrase? That's the part that seems odd to me. I would either make it all a quote (using ... to indicate that you left out words or more material, or using to put in clarifying words that weren't in the original quote) or make it all a paraphrase.

My preference would be to see it all be in your own words. Why? Because these are simple facts. There may be different numbers out there so the exact numbers you use might be disputed. But it's not an opinion really, and it's not anything you need to quote.

In either case, either use a footnote for the citation for the entire sentence, or use a combo of a footnote plus words in the text.

For example:

BitTorrent traffic accounts for 18% of all internet traffic, two thirds of which is estimated to be illegally distributed copyrighted material.[3]

OR

Maria Gomez estimates that 18% of all internet traffic comes from BitTorrent. And that two thirds of that is illegally distributed copyrighted material.[3]

In both cases:
[3] Gomez, M. (2018). BitTorrent Statistics. Journal of Internet Trivia, Vol. 9 (4), pp. 105-116.

While you don't have to name the fictional Gomez, I like the active voice version much better.