The Star Rating Promotion content for mount will be opened when the sub-quest is completed after reaching character level 330.
You can use the content through [Mount] > [Promotion] in the lower right corner of the game.
You must own the mount you want to promote in order to proceed the Star Rating Promotion using the mount summoner item. Also, you can get your mount promoted using Dragon Orbs that match the tier.
Mount Forge content will open when the star rating reaches 10 through star promotion.