It did for both my mining quests at Skull Valley today. At first I thought it was something in the room clicking away, but i turned up the volume on the TV and there it was. It stopped as soon as I started mining, then started up when I put my pickaxe away.
There were three sounds. The ticking of a clock, my footsteps and the BGM.
I never noticed it before today. So either there was a change with the latest update, it's been there the entire time since launch or I've finally gone insane...