Yes, the words have a common root. The question is perhaps a bit confusing, since there are several very similar, but unrelated word in the Swedish language.
The first word, has as mentioned in the comments to the question, probably an onomatopoeic origin and is a cognate to English 'whine' or German 'weinen'. This is not the word, which is asked about in the question.
In the second meaning, it is an alternative spelling of 'att vinda', a directly related verb form of 'vind' (wind) in the meaning 'what the wind is doing'. I would say that the usage is slightly exalted and not common. The usual expression would be to say that 'vinden blåser om hösten' with 'blåser' as a cognate to 'blow' or 'blasen'. This is the word used in the mentioned sentence.
We must also be careful not to confuse the previous meaning of 'att vinda' with yet another unrelated word 'att vinda', a cognate to 'wind' or 'winden' in the meaning turn or wrap around. Even if very similar in all three languages, this word actually has a different etymology than the word for moving air.
Then we have the German word 'wehen', which is cognate to Swedish 'vaja'. I first thought that we here may have English cognates in 'wave' or 'waft', but those words seem to have different origins. In Swedish, the word has a more restricted meaning than in German. In German, the word can both have an active meaning 'der Wind weht' (the wind is blowing) and a passive meaning as in 'die Flagge weht im Wind' (the flag is blowing in the wind) as in the action imposed upon the flag by the wind. In Swedish, the word can only be used in the passive meaning as in 'flaggan vajer i vinden' and not in the active meaning. A Swedish sentence like 'vinden vajar' would not make sense.
And then back to your actual question: The words 'vinda', 'vind' und 'vaja' (and therefore also German 'Wind' and 'wehen') can according to 'Svenska akademiens ordbok' be traced back to Sanskrit 'vāti' (to blow) and have a common origin.