No, you're not obliged to meet with the drunken mayor of Glenborus to get the Axe of Stone quest finished. In fact, I completed the game three times and had never talked to the guy about it. If you've talked to Eolus and the imprisoned elf, that's all you should need to get the dwarven mines opened up.
Although, I can't remember exactly how you're supposed to know to go to the dwarven mines...maybe I just read a quest guide.
