In a word, no. Laptop or desktop computer usually have USB ports, which put out like 10 watts of power for a "high current" USB port. That is not enough to do much heating or cooling. You want a high power supply. The pelteir plates I've seen run on 12V DC. Get a 5A 12V supply, or even higher.
You might be able to draw larger amounts of current from a USB-C port or a USB-C power supply, but why? Get a dedicated high current power supply that puts out the maximum voltage your peltier plates can handle.
Note that there is no reason to have 2 plates for heating and 2 plates for cooling. Run the current one way and they heat. Reverse the current and they cool.
Also I'm not sure of the value of using 2 plates to cool. A single plate able to handle more power would likely be more efficient, and easier to insulate. (With 2 plates you'd need 2 heatsinks, which will allow twice as much heat to leak through when the power is off. Doubling the current you feed into a single plate should give double he heating/cooling, without increasing the heat leakage when it's off. Just make sure you don't exceed the maximum rated current of your plate, and that you have a large enough heatsink to handle the heat transfer.)
You could use an H-bridge meant for controlling motors to provide either +12V or -12V under software control. Those generally handle high current.
You could also use a digital thermometer and make a thermostatically controlled system. You'd measure the temp in the bottle, and when it got to your desired temp range, turn off the power. When the temp moved outside the desired range, you'd turn the power back on.