So you want the Chevy bowtie to be the quilting design inside of the blocks. Binding, as ckcowl explained, is the fabric you use all around the final edge to hide and contain the batting and final edges of the quilt.
Yes, almost any longarmer should be able to find or create a Chevy bowtie to quilt inside your blocks. The possible problems I foresee are this would be considered a "custom" design is will cost you more. Also, it will be a stop & start in each block which will also cost more unless they can figure out how to wiggle or stitch the way unseen or at least without detracting from the bowties in each block. Lastly, the silver thread. If a metallic is used--many won't even use metallics--it will cost more but if a nice silvery gray cotton is used instead, it won't cost much more if any than standard cotton quilting thread. There are many lustrous cotton threads out now that will behave much better than a metallic.
How large is this quilt? I would definitely PM Charismah or any of the popular longarmers on this board & get their professional opinion. I only quilt for myself at this point.