After you accept the quest, if you already have Basalt Grotto on your map, it will be faster to map travel there and exit back into the Mines. If you do not already have Basalt Grotto simply work your way north-west through the groups of Margonites and Kournans, following the quest marker as usual. If you follow the latter suggestion it would then also be advised to zone into Basalt Grotto during the quest so that you do have it mapped.