If you came here from Google with the question:
How to connect a 4-pin (3-pin) connector to a power supply?
This is for you. The OP asked
I currently have a fan with a 4-pin connector that I want to power, but I do not have a power supply with a 4 or 8-pin connection. Is it possible to disconnect this connector piece and connect the individual wires directly? (red and black?)
I interpret this as "I want to power my fan and have it running at 100% speed".
If you interpret this differently, ask the OP to clarify. Otherwise:
Is it possible to disconnect this connector piece and connect the individual wires directly? (red and black?)
Do you have red and black wires? I don't. I have four black wires on my 4-pin PWM Arctic and Noctura fans because I use my own PWM cables. So, to answer the OP's question and help Googlers years in the future, try this:
Attach a voltmeter to any two adjacent pins/wires (power pins are usually adjacent). Next, with a can of compressed air (not by touching or blowing), get the fan spinning and see if you can observe something close to the 5V, 12V, or 24V (what your fan is rated for). You should only need to try this at most twice. Yes, this works.
Why does this work? If you power a typical (single-phase) DC cooling fan, it spins. If you blow on a fan hard, it produces a voltage. This is how you find your power pins for the 3-pin and 4-pin PWM fans if you want to run them at 100% duty (no PWM signal at all - a floating pin). Nitpickers will nitpick, but this has worked for ages for cooling fans in computers and 3d printers.
You can even pulse the power wires (it is still considered PWM) for low-power fans. This is exactly how 3d printers control their mainboard cooling fan (e.g. BTT SKR mini E3 v3). Although, adding a capacitor across the power lines with a PWM signal is cleaner, but that is another topic.
That is how you 1) find the power pins for a 4-pin PWM fan, and 2) yes, you can power the fan for 100% speed with just two of four pins.