Too Long; Didn't Read (tl;dr)
That circuit was designed for 40 m. You can't modify it just by replacing the quartz crystal to work on 2 m.
What Would Need to Change?
So what you've found is a circuit designed for the 40 m band – you can use the same general principle to build a circuit for 144 MHz, but you can't use the same circuit.
- I've not seen a 144 MHz (in fundamental mode) crystal yet. You might have a source for these – they can be ordered at custom frequencies. I will say that going to a fundamental frequency that high stretches the things you can easily do with the mechanically oscillating piece of piezoelectric crystal quite significantly! So, this might simply be where "using a crystal oscillator's fundamental frequency of oscillation" dies.
So, you need to redesign the X1/C3/L1/Q1 circuit completely to allow for operation at a subharmonic of 144 MHz; you'd then probably want another active stage and filters to generate a clean 144 MHz harmonic. So, fundamental design change!
- the 2n3904 is a real transistor. While it does have a sufficiently high speed to theoretically work as the active part in an oscillator at 144 MHz, you need a very specific operational point (in terms of base-emitter voltage and collector source impedance) than this circuit has.
So, you need to redesign the whole crystal oscillator subcircuit. While you're doing that, you'd replace the transistor with a different one. But you're already doing that because of the previous bullet point.
- Because you're now working at a different target frequency and in a non-fundamental mode, the C6-C8/L2 matching-and-filtering circuit needs to be redone. Since you mustn't leak any subharmonics of 144 MHz, a low-pass filter won't do – you need a bandpass. Your antenna might pose part of that, but it really depends on your antenna design, and one your 144 MHz synthesizer design.
So, all in all, what you found is a circuit. Circuits are always designed with the reality of how their components exist and work in mind. Going up in frequency by a factor of 20 typically means that these design assumptions don't hold anymore – and you thus need to replace significant parts of the circuit.
Circuit "Idea" Critique
Regarding your original circuit: I do feel a bit uneasy looking at it. It's a fine oscillator, with probably a fairly clean output spectrum (i.e., a single tone at the desired frequency, with the width of that tone dominated by the quality of the quartz) as far as I can tell (really not a low-frequency oscillator person myself), but you'll probably find the buzzer significantly modulating your tone, and you'd usually want as clean a tone for CW as possible.
I probably would have split this design for 40m, already: Instead of having a single transistor that does both, being the active part of the oscillator and the source of power going to the antenna, I'd have an oscillator circuit, which feeds into the amplifier circuit, which is what your key turns on and off. Notice how the author says in their v1 of the transmitter the transistor was overheating – that means they were operating it at a power loss in the transistor higher than what the transistor can deal with (and the onsemi datasheet p. 1 says that's 625 mW (!)), which would have shifted time it takes for the transistor to turn on by orders of magnitude (fig. 5 on p.4), so it would have severely detuned the oscillator. The solution to this is (at these low powers) to not behave like the transistor is the expensive part here¹ , but to have one transistor² in the oscillator, operated exactly in a point that's optimal for low-power, high-accuracy oscillation, and a different transistor³ acting as a power amplifier, operating at exactly the point that's optimal for that. The added bonus to this increased modularity is that you can let the oscillator run continously – meaning that there's no weird turn-on-and-warm-up-until-frequency-stabilizes behaviour every time you press the key. (As an engineer: If you can build a system modularly, you do it, because that allows you to measure, adjust, fix, exchange and generally optimize each part individually, instead of always having to re-tune the whole system as soon as you change the smallest part of it. Imagine a gasoline car where the amount of current your turn indicator takes influenced the timing of the ignition sparks. Engineers would go insane trying to get that car road-safe!)
In this light, I would, especially since you're eager to build things yourself, and improve on existing designs, propose the following:
- Start with a ready-made oscillator (instead of building an oscillator from a transistor + quartz resonator). That way, you know you have a reliable oscillation, and can design the antenna amplifier circuit.
Yes, you can (and maybe you should!) replace that ready-made oscillator in a second iteration of your design, with something that you make from a quartz resonator such as X1 in the circuit. Nothing wrong with building a working prototype where you don't solve all problems at once!
- Since 144 MHz oscillators are rare (exactly because 144 MHz crystal resonators are very hard to produce), you'll end up with an oscillator that works at an odd divisor of 144 MHz; 28.8 MHz (= 144 MHz / 5) is relatively common. So, all you need to do is get the fifth harmonic of your 28.8 MHz oscillator.
If you picked an oscillator that has a square wave output (instead of a sinusoidal output), all you need to do is build an RLC band-pass filter knock out the fundamental (28.8 MHz), the third harmonic (3·28.8 MHz = 86.4 MHz) and the seventh harmonic (7·28.8 MHz = 201.6 MHz). The ninth and higher harmonics will probably be already sufficiently attenuated.
If you don't have square output, a simple logic gate (like an inverter from a 7400-series logic IC) might be sufficient to convert a sine wave to a square wave, introducing exactly these harmonics you need).
- Whether you place the power amplifier (to get to 10 mW or wherever you consider QRP to be) before or after the filtering stage depends on what you want to achieve. Personally, if my goal was minimal complexity, and relatively low losses, I'd probably put it before the filtering stage, and make sure I have a high-Q bandpass filter. This has to do with why the original transistor got hot: it was operated in a situation where it was not "fully on" nor "fully off", most of the time (basically, all of the time). But: our oscillator gives us a square wave; we can be fully on or fully off, nearly all of the time (the times when we're not fully on is during rise time and fall time of the square wave's edges).
- Designing the filter depends a bit on where you put it – it'll also transform impedances. But, for our requirements (letting through 144 MHz, passband ripple nearly doesn't matter, stopping 86 MHz with > 40 dB and 201 MHz with at least 30 dB), a simple third order Chebychev bandpass filter would work, or a third order Cauer/Elliptic filter (really depends on how much you want cleanliness vs how much you're inclined to hand-tune your filter circuit; rule of thumb: the better a filter the more fiddly you need about tuning it in the face of component values that vary in reality)
¹ this being a single transistor design betrays the 1950s/1960s heritage of the circuit idea, when transistors actually were more expensive than the rest of the circuit. Nowadays, having as few transistors as possible doesn't make the circuit cheaper or easier – it makes the circuit harder to tune and needs more components for compensation of unwanted effects!
² or some more complex inverting active element
³ or some more complex amplifying active element