First of all there is geneally three or four divisions of LAQ. The cheapest cost is what's called an "All Over" quilting, where the quilter just does a meandering pattern all over the quilt. Depending upon the area in which you live, this usually costs around a penny a square inche.
Then we have the panagram patterns, in this the quilter follows a pattern with a laser laid down on a sheet of plastic along the machine table. This usually cost a oenny to a penny and a half per square inche.
Then there is "Free Motion" quilting, where the quilter quilts specifiic agreed upon patterns on different places of the quilt. This can run from a penny and ahalf to 3 or 4 cents per square inche. An example of this would be where features are sewn around a quilt and McTavishing or Stipuling is used in other areas.
The method that is best for you depends upon what the quilt is intended for. If its for a child to drag around, you would probally use the all over method. If for a gift, you would probally use a more expensive quilting.
As far as what to ask, its always good to see an example of their work. Also ask for references that have used them.
Get a cost proposal from them. This can always vary a little bit, but should be fairly close.
Make sure that they know exactly what you expect them to do, and to contact you if their is anything that comes up that impeds that.
What the time frame is when they can reasonally get it back to you.
Those will help you decide on who quilts your quilt. After putting so much of yourself into a quilt, the quilting should enhance your work, not overpower it.
By the way, a good quilter wants you to ask these questions so that there won't be any misunderstanding.
Hope that that helps.
Lanny