hello everyone; I have a small external speaker with its own electronic board which has a 12 volt power outlet and an external speaker output as shown in the attached pictures, The board has LM324M amplifier, My question here is can I use this speaker as an external speaker for the computer using an AUX cable

 

The LM324 is just a low power op amp, and has little to do with your question.
What are the connectors on your AUX cable?

Is the only audio input to the amp, the MIC socket?
If so then, if your computer has an audio output, you may be able to connect that to the MIC input of the speaker amp to drive the speaker.
But you will likely have to attenuate the signal with a two resistor voltage divider to reduce the signal level and avoid overdriving the MIC input.
I would suggest starting with 10kΩ in series, with 1kΩ resistor to ground at the MIC input.

 

The much larger TDA1905 is the power amplifier. The tiny LM324 is a preamp.
The LM324 is never used in a hifi audio amplifier because it has hiss noise, crossover distortion and a poor high frequency response but here it was used as a telephone amplifier where its bad sound does not matter.
Made for a telephone it might not work properly from a computer.

Buy some stereo computer speakers. Mine sound almost hifi and sound so good that I do not bother to switch my computer sounds to my nearby hifi.