Is there any way to convert a Photoshop Style file (*.asl) to an Illustrator style file ?
This is a very unclear question, so here is a pretty wild guess; you want to change raster image to vector image?
Raster image uses pixels, gets blurry/blocky when zoomed. Vector image uses math, can be zoomed infinitely. Some images (logos, flat colors) can easily be made from raster to vector. But complex mages (photos) are very hard to convert, and you should hire an artist.
There are tools, for example https://www.vectorizer.io that can try to convert raster to vector, but results depend on the style and quality of your picture.