Is it possible to use a 2.4GHz magnetron as a oscillator for a Tesla coil? The short answer is no, definitely not.
The long answer is... well, long. Note that the following reasons are each, by themselves, absolute show stoppers. Only one of them would need to be true to make a 2.4 ghz Tesla coil totally impractical. Unfortunately for this idea, the situation is much worse than just one of them being true: all of them are true.
How would you build a 2.4GHz resonant transformer? Remember, that's ultimately what Tesla coils are - air-cored resonant transformers with relatively weak coupling (around 0.2 or less. You can't just hook whatever frequency you want up to one, you need to make sure the primary will resonate at the same frequency of your oscillator. Typically this is done via an LC tank, but the capacitance used is the parasitic capacitance of the coil, so there is no lump capacitor component added.
So, just making some numbers up, if you wanted a resonant frequency of 2.24GHz, not even reaching our desired 2.4GHz, if you had 5 pF of parasitic capacitance, which is an unrealistically low number. These leaves you a whopping 1 nanohenry of inductance for your coil. This is the inductance of a wire roughly 2.5mm long. These are the scales you have to work with, and factor in how difficult it will be to construct a 2.4GHz resonant transformer that has even a remotely usable primary-to-secondary capacitance ratio (it is this ratio that determines the output voltage of the secondary, not the winding ratio like in non-resonate transformers), just, well, it isn't practical. The reason Tesla coils step up voltage at all is that by using the secondary winding's parasitic capacitance, and using a relatively larger (several hundred pF or more) tank capacitor in the primary, we get this large capacitance ratio, and after that, the voltage step-up is simply a matter of conservation of energy.
At 2.4GHz, you no longer have any practical way to achieve the needed ratio of capacitances between the primary and secondary. Even if you use fractional turns, not even a single complete turn on either coil. The physical dimensions and parasitics simply do not allow for the construction of a resonant transformer at that frequency that has any significant voltage step-up capability. Which leads me into..
The maximum output voltage of a Tesla coil is more or less inversely proportional to its resonant frequency. Let's be extra clear about this: the faster your coil oscillates, the lower the output voltage. The main reason to use higher voltage is it allows you to build a more compact Tesla coil, but the ultimate output voltage is proportionally lower. Higher voltages requires physically larger coils, which require lower frequency operation. The highest frequency Tesla coil I've seen was 4MHz, and it was comically tiny. Imagine a Tesla coil for ants. For all intents and purposes, any Tesla coil that is interesting enough to even be worth building is going to be under 1MHz, with the really huge ones operating at 20kHz or even less. The lower frequencies permit wider and wider primary to secondary stored energy ratios, and thus higher voltages.
Simply put, anything you might build that is operating at a resonant frequency of 2.4GHz would not be called a Tesla coil by anyone - including Tesla. It would achieve no useful voltage ratio at all, if it could even be made to be gin with. The most spectacular thing about it would be how spectacularly it vaporizes from the magnetron's 1800W of 2.4GHz electromagnetic murder being dumped into a resonant transformer the size of half a grain of rice. Might be as much fun as a tantalum capacitor hooked up backwards, you never know!
Electricity doesn't really behave the same at 2.4GHz. You can't just send it down wires anymore. It essentially doesn't even conduct as current, as for copper, the skin depth at 2.4GHz is 1.33 μm. At that point, it's more useful to simply treat it as an electromagnetic wave - as conductors will see little more than tiny surfaces currents that just radiate the energy back out again. You definitely can't just hook a wire up to the business end of a magnetron, as it is set up to radiate electromagnetic waves. It is not producing 2.4GHz AC current, it is a cavity resonator and amplifier that produces microwaves by accelerating free electrons back and fourth - there is no actual conductive path for the electrons to go anywhere, and if there was, it would change the resonant cavity. No, you're getting pure electromagnetic waves, and now your primary coil is forced to be a incredibly beefy wifi antenna if it want's to pick up any useful amount of the radiated energy in a controlled way (as usable AC current).
Good luck with that. You know why metal inside a microwave oven sparks? Because the electric fields generated from those 1800W of microwaves (or whatever, varies but usually over 1kW) are so intense that they will produce a voltage potential across even tiny bits of metal that are large enough to cause dielectric breakdown of the air. In other words, pretty much the worst possible thing you could have near ANY Tesla Coil. Or anything metal. Or people.
At 2.4GHz, you'll have an enormously difficult time just bottling up the energy. You have horrible losses due to just about everything acting like an antenna and radiating out all the energy. And insulators are fantastic conductors of electromagnetic waves, while conductors block/reflect them. Sort of the opposite of actual electric current. The thing is, any time you accelerate a charge carrier, it produces an electromagnetic wave. And when you're current is wiggling charges back and forth at 2.4GHz... well, you start losing energy to those generated waves a lot more severely. It makes even the simplest things hard. You have to use wave guides (coaxial cables) instead of wires...on and on. But I digress.
Oh, and on a related note, you should not power a bare magnetron up, and you are definitely not capable of doing this safely if you are even considering doing it in the first place. It takes a lot of experience and theory and even then, predicting how microwaves will behave is difficult. You need to shield anything near by properly, and it is very easy for a large amount of RF energy to concentrate in human tissue and cause internal burns that we lack any nerve or pain receptors to feel. Expect to cook an organ (how many non-vital organs do you have to be cooked? Not many) or, the most common injury, is simply destroying your retinas and you just don't get to see things anymore for the rest of your life.
And even if you aren't concerned with your own safety, it is unlikely you'll be able to keep nearby pets or people sufficiently safe. Just don't play with magnetrons, for everyone's sake.
Also, most magnetrons are dangerous if they are simply exposed. Does yours have a little bit of purple ceramic looking stuff near the antenna? That is beryllium oxide. In it's bulk ceramic form, it is fairly safe, but like any ceramic, it tends to form very fine dust particles in the event it breaks or is crushed. You do not want to breath those in. They can cause Berylliosis, and as a bonus, they're carcinogenic to boot. This isn't some OSHA level, cover your butt safety crap - you really legitimately do not want to mess around with stuff like this.