Yes... but No
Okay - bold statement here, definitely not hedging my bets...
So - in terms of the basics - yes, an instrument with a constant flow of air could work.
You might have some technical challenges around things such as needing to 'tune' the instrument to account for the pitch/timbre of the jet flow as is - but that is doable.
Donning my musician hat
If we are just looking at the woodwind/brass section, we have a number of problems:
Sometimes phasing is used to cover up the fact that we humans can't expel our breath infinitely (quick aside to trumpet players and cheek puffing) - but phrasing is also used in a musical context - let's think of the Baker Street sax solo - the first opening phrase is a bold statement, tailing off to the second phrase which reprises the first theme (linking it) but changes the second.
The fade in/out of the phrasing is what makes it sounds musical. Now, granted you could overcome this by having some form of valve system to divert gasses away - but it's still an issue.
A good musician will have a wide dynamics range - again, woodwind and brass section considered here - the ability to go from piano to fortissimo is a skill. A jet engine is very good at sitting at a constant speed - and whilst they can spool up and down relatively quickly, They would not be able to go from quiet to loud quickly.
You could, again, perhaps fix this with a complex series of valves...
For some (not all) woodwind/brass instruments, pitch is a function of the valves and also of the musician themselves - being able to go from high to low and do so quickly - for reference here: This is an 'experiment' in DCS world of getting jet engines to spool up - the quickest being 3.something seconds - That's half a 4/4 Bar at 100 BPM. And again - that's the quickest most being 5-6 seconds, so a full Bar. Not only that, but changing the blade RPM (which will change the pitch up and down) will also increase your dynamics - so trying to play high and quiet is an issue.
Settle down. This is a legitimate musical technique - where you use your tongue to interrupt the airflow to play 16th note or even 32nd note patterns. Think of the William Tell Overture - 'da dadada dadadadad' - that repeated note is double tongued.
Again - possible with valves etc., but good grief we are getting a lot here.
Instruments don't like getting hot or cold, this is why things like glass flutes used to be popular, because they wouldn't get out-of-tune when they heated up or cooled down - a jet engine produces a lot of heat - that heat is going to be doing some expanding of the pipes if it is made of anything metal (probably can't use wood... Ceramics might work) - and that will do all kinds of weird things to your pitch.
This is perhaps a summation of all of the above - but there is more to playing a woodwind or brass instrument that just a constant flow of air, I'm a drummer myself (although did learn piano and can remember a lot of musical theory) - there is so much nuance in how a musician uses their body to play the notes. The speed of transition of the human body is much faster than the speed at which a jet engine (even a small one) can spool up or down - and to overcome all of these challenges would require so much extra valving, it would be incredibly impractical.
A jet engine would be great at providing a drone (like the bagpipes) or a siren - or anything that didn't require changes in pitch or tone or timbre or alike.
That said - I like airplanes, I like jet engines - and so if you want a world where there are jet engine instruments - you could do it - the music would have to be written specifically for the limitations of the instrument - but you could do it.
To conclude - instruments as we know them and music as we know it now - no, a jet engine would not work. But in terms of a raw air-source to make different kinds of music suited to the above limitations - sure, why not
Edit: If Tchaicovsky can use a cannon as an instrument for the 1812 Overture, then you can use a jet engine.