It's helpful to distinguish between some different interpretations of the word 'key' here:
If you're referring to the letter name then there's really nothing all that different between the sound of, say, a piece in Eb and that same piece in C#, relatively speaking (at least on modern, equal-tempered instruments—this was not necessarily the case throughout all of history, however I'll spare you the technical discussion as in today's world the use of non-equal temperaments is quite limited). The intervals will be the same, the intervals should be equally consonant or dissonant in any key. There can be some differences on certain instruments, say a cello with open strings which may ring sympathetically in certain keys, or a trumpet, which might sound shrill when transposed to a key that necessitates a higher range, but on the piano these differences will be pretty negligible.
If, instead, you're referring to the tonality of a piece, minor or major for example, then the answer may indeed be yes. We've inherited a great many emotional associations from our culture's long musical history, such as the "minor is sad"/"major is happy" sort of association that most people recognize. It's important to note, though, that the mood of a piece depends on so much more than just its tonality—things like dynamics, tempo, articulation, and so on can elevate or depress a mood totally independent of the piece's tonality. One of my favorite forlorn ballads, Hoagy Carmichael's "Skylark", spends most of its time in a major key, but to my ear expresses a really lovely, heartbreaking kind of sentiment. Conversely, some performances of Arthur Schwartz's "Alone Together" render it very upbeat, despite its predominantly minor key. It's also important to note that these things are really subjective, and one person may get a wholly different perception of a piece than the next.
So to summarize my answer here, there isn't really a clear answer except to say that your job as the composer is to choose a feeling you wish to express and then make other decisions about the music in support of that feeling. Things like the key of a piece are absolutely a part of that decision-making process, and can be informed by our cultural history. Listen to and analyze as much music as you can, trust your ears and your heart, and you'll find some clarity on this point.