If you are using the statistical software R you can use the R package fitdistrplus and the function fitdist. For example, if your variable is called x then use the code fitdist(x, "weibull") and it will provide you with the estimate and the standard deviation for the shape and the scale. You can also plot this function to see how the diagnostics looks and if the distribution fits your data.

I outlined two answers to the questipn you are asking:

1. Method of moments:

https://www.researchgate.net/post/Can-anybody-explain-how-to-find-out-the-shape-and-scale-parameters-for-weibull-statistics-for-average-wind-speed-data-for-a-month

2. Least squares

https://www.researchgate.net/post/How_can_I_determine_weibull_parameters_from_data

Good luck.

Yes, follow Carl Alexander Frisk instructions. I did that for my PhD :)

Thank you everyone!