A quick and easy way to produce a quilt is to try a "Rag Quilt." These are made from flannel or homespun fabric. Fabric squares are machine quilted as part of the construction and when the squares are joined together--you have a finished quilt! The wrong edges of the fabric are visible on the top of the quilt and are clipped to encourage fraying, further adding to the "rag" look. Check this link for a pattern from JoAnn.com.
If you need a quick project for a gift, try a quilt of large squares. One of my favorite project was a large lap-sized quilt made of 8.5 inch squares that was backed with flannel and machine quilted. If this is your first quilt, you might want to skip the flannel and use a smooth cotton for the backing. Flannel can stretch if you aren't diligent in basting.
Here's a photo of a small section of my lap quilt.