Take a deep breath and relax: mistakes happen all the time, and most of the time they do not matter.
The only really important questions here are:
- Does the mistake change the meaning of the quotation?
- Is the mistake worth correcting?
If the mistake doesn't substantially change the meaning of the quotation, then it's almost certainly not a problem. For example, if a typo or transcription error caused a word to be duplicated, mangled, or dropped, then it may be clear that there's an error, but the intention of the communication will be clear. An easy example generating this is when a copy-paste runs into problems with ligatures or text selection order. If it makes it past not just you but also some peer reviewers or copy editors, then clearly it wasn't a big deal, and nobody will hold it against you.
Even if it does change the meaning, you are presenting the material in some interpretive context, which should make clear what the quote was intended to contain. Again, if you aren't distorting or misrepresenting the actual contents, you should be fine.
Now, as to whether it's worth correcting: my general opinion is that it's worth correcting if the mistake changes the meaning, or if you just can't live with incorrectness once you've noticed it. Issuing a correction is often a hassle, however, so if it's just an inconsequential typo, I would let it stand.