Looks like the nut is pretty worn, check 1st fret action across all strings? Any hidden cracks in the wood? Have you checked for loose screws, either mounting the tuners, the tuning gears or the tuning pegs themselves? I'm more of an electric guy, but I'd start there and see if the problem reveals itself.
I agree about the nut. Also, the windings look pretty wonky and uneven. Are they making contact with the cutout in the headstock before they go onto the nut? Might be causing some vibration there.
Where the posts around which you wrap the strings sit. I dunno if it has a particular name, but in woodworking, those two oblong "windows" on either side of the headstock would be cutouts.
It looks like your windings on the top and bottom strings might be making contact with the sides of those cutouts, and that could be causing some unwanted vibration.
Ideally, all of your strings would have nice coils that wind across the post (towards the middle, in this case), wrapping in tight coils that touch. They should also wind a certain number of times, but I dunno what the recommended number of windings is for a classical guitar. Typical guitars are 2-3 for the heavier strings and 3-5 for the lighter strings. Look up some guides on how to correctly string a classical guitar.
I had a similar thought, but I don't think so. I'm pretty sure they're solid throughout. It's why you have to be careful not to overtune them, because they will stretch too much and be permanently deformed.
If you have a capo, you could test it on the first fret (or second, or third, etc.) and see if your problem stops. I had a guitar with a worn nut that put a couple strings just low enough to always buzz against the first fret.
The ends of the bridge knots. These should have been tied so that they are looped into the knots of the other strings, but if they are sticking out and touching the soundboard you may get some vibration.
Same of the ends of the strings around the tuning pegs. It looks like these have been trimmed so that there isn't any extra string after the knot around the tuning peg, but if there is it will sometimes touch the headstock and buzz (I don't trim mine, so this happens to me sometimes).
Someone else mentioned that your strings are touching the wood due to how they are wrapped - also a possibility. I'd loosen the strings and push those wraps further onto the post.
And just a wild guess - were you playing while wearing a shirt with buttons?