With instruments that old there might be many reasons: action too low, uneven frets, degraded nut... The best thing you can do is bring it to a professional luthier. A good one can find what's your problem in 2 minutes and fix it.
Since that's a classical guitar something you can try for free is tightening the tuner pegs: there should be a screw holding the gear wheel and if it's loose try to tighten it with a screwdriver
BE GENTLE, DON'T OVERDO IT.
If you're lucky it might be it, otherwhise do as I said earlier and bring it to a luthier.
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.
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?
Seems like someone filed down the nut for lower action but might have gone too deep or made the channel too wide. If it sounds like you're playing a sitar it's probably the latter.
The plate that the tuning pegs sit on looks loose at the top right of the picture, just above the peg for the g string (heh heh, g string 😁 ). Try playing the g string and holding the plate in place.
If it's not that, try putting your finger on the other parts while you're playing, one at a time until the noise stops.