Yes, the article a can be used to precede the noun wind. However, this is a special case. For nearly all constructions, you should prefer the wind. As in, "The wind was blowing when I went to the store," or, "The wind carried my hat away."
As you stated, using the article a before wind generally occurs when you are modifying the word wind. A gentle wind, a strong wind, and a warm wind are common. You could use a wind by itself, such as, "Boy, there's a wind blowing through here tonight!" This uncommon phrasing brings emphasis to the wind, and makes the sentence stand out more than if you were to say, "The wind is blowing tonight."
In most cases, however, you should use the wind.