Should the contractor be able to drill holes in a quartz countertop?
Sink holes (faucet/soap/whatever) are almost always done on site.  The actual sink cutout is already done at the shop and the company doing the install probably does not want to risk even further the chance of cracking/breaking.  

Anyone with a drill and a $15 diamond circle bit can cut these holes out in a couple mins per hole.  If they can't they shouldn't be installing your countertops.  

If your contractor says they are a pro at installing countertops and they DO NOT do this then it is equivalent to a plumber saying they don't hook up shower heads - get another installer.  The holes should be drilled before it is on your cabinets (often these holes might not leave much room from behind for a drill to go straight down).  You do not want to have another guy come out and drill these - if something went wrong who would be at fault?