Also realize that if you have a Pantone Swatch Book and it is several years old it is no longer accurate. Ad agencies and printers generally replace their books yearly.
Coated stock (gloss) and uncoated (matte) are quite different in color saturation too.
Color matching is not an easy task unless staying within a coloring system like Pantone from start to finish.
It would be better for your client to supply a sample of what color they want to match and then eyeballing the sample against your mat board samples to come up to the closest match. In lieu of that, you might be able to have a printer use a Pantone printer's ink to print a board to match exactly prior to cutting the mat. I don't know if the Pantone papers are still made, but do not use them to mount on a mat board because they too will fade relatively quickly as they are made for mock up use only.