English isn't written with Chinese characters, so generally you cannot write "swan" as 鴻. If you're talking about assigning meanings to characters (from a Japanese standpoint), then of course you're a bit more free, but ultimately this is a bit of an arbitrary task.
That being said, my Kanji dictionary, in the meaning section, lists "big water birds" and gives クグイ as an example, which is an archaic word for 白鳥 (swan). Notice that the opposite is not necessarily true: If you look up a Kanji spelling for クグイ, you'd end up with 鵠 and not 鴻. But that's not surprising, as the meanings are inferred from words (compounds etc.) that 鴻 is being used in and don't tell you whether the character is used by itself stand-alone in any capacity.
Most common compounds I can find in my Kanji dictionaries with this word seem to only use it in the meaning "big, large" (as an opposite of 雁).
meaning
Can we use this kanji for "swan" too?