No, it's seven.
Location and setting
This is a good question, and it shows that you are paying attention to details. If you keep looking attentively, you'll find that these are two segments from the same longer passage in the same setting, both describing seven spirits in the same place: near "the throne" (τοῦ θρόνου).
Both times, it is "the seven spirits" (τῶν ἑπτὰ πνευμάτων - 1:4 genitive plural; τὰ ἑπτὰ πνεύματα - 5:6 nominative plural). It's not "seven spirits" nor "seven other spirits" nor "seven more spirits", but "the seven spirits". This definite article adds a bit of clarification in the Greek.
Generally speaking, we can't always be sure whether there was a definite article in Greek just by reading English because sometimes Greek uses the article for the same purposes, sometimes for other purposes. In this situation, Greek uses the definite article "the" (τοῦ genitive singular, τῶν genitive plural,τὰ nominative plural) almost exactly the same as we would use "the" in English.
Not to condescend, because this is a good question, but it is good to look at how literature works. To conclude that there are fourteen spirits because "the" seven are mentioned in the same place twice would be similar to saying that a boy has two mothers simply because he said, "my mother," in two separate sentences. It might not seem that simple, but when we become familiar with the passage—probably just by reading it a lot, though Greek can help—it really is that simple.
Great question! I hope that is helpful to understand not just this passage, but Greek articles and literature in general as well.
As a side note on how that relates to the idea of "the [singular] Holy Spirit", that would be a separate question—how to read these passages in light of each other. For example, Revelation could give a closer look at different aspects of the same Holy Spirit, or something like that, but that's not our purpose here. For this question of whether Revelation is telling us "seven" or "fourteen", the answer is a clear "seven" given twice.