The first question is whether the Weibull distribution is still a good model of wind speed when it is a monthly average. I'm not familiar with this field, but from what you say it sounds as though average hourly wind speed is often modelled as having a Weibull distribution. If you take the average of 700 or so such random variables (24*30) the distribution will be very nearly normally distributed because of the central limit theorem, even with the autocorrelation of the underlying hourly observations.
I'd suggest looking at the actual distribution of your 60 data points and comparing it to a normal distribution, using something like the qqnorm() function in R to draw a plot being the obvious starting point.
But basically, I doubt the probability distribution of your variables is your main problem here. The challenge with modelling your data will be more related to the need for you to estimate and control for seasonal effects from quite a small data set (by time series standards). Exactly what sort of challenge that is though depends on what your research question is (for example, do you need to see if one year's wind speed was different from others? or are you looking for a trend? or what?).