I have seen pieced quilts done on a serger. Tried a couple myself. The problem with using a serger for piecing is that it's just not very accurate. If using a serger, typically you want to do a quilt that has very large pieces and/or do as much piecing as you can in long strip sets. Also, the serger adds a lot more thread to the seams than a sewing machine does, making the seams much more bulky. Although it's possible to piece quilts on a serger, almost all quilters prefer using a sewing machine.
The same problems apply to attaching borders and binding. It is not easy to achieve accuracy, and the seams are considerably bulkier.
I don't even like to serge the edges of a quilt before binding, which is probably the most common use of a serger by quilters. I did not like the stiffness it added to the edge of the quilt.