Step 1, train your mage in deception
The funny thing about chronic gossipers like your mage is that people who are bad at keeping secrets tend to also be good at telling lies. Give him a cover story so that when his friends ask him what he's been up to, he has a brag worthy lie to tell in place of the truth. "I've been working on new methods of enchanting armor! but shhh, don't tell anyone because it's a secret." Lies in place of the truth are much easier for gossipers to maintain than silence; so, in this way your mage will be able to keep things under wraps for a while, while still having something to brag about. Now that said, eventually the truth, or part of it will come out; so, when it does your mage is also trained to lie about WHEN he discovered the thing, and report his slipup directly to the queen right away. So if he accidentally slips and says he's been making gold, he can immediately back peddle and say that it was something he accidentally discovered last week while doing his armor enchanting work.
Step 2, exploit the discovery while creating plausible deniability
Dumping your gold into purchases or going into debt are great ideas already listed, but as Otkin and user253751 have pointed out, this kind of insider trading will be horrible for your foreign relations. This means that on top of doing insider trading, you also need a really good scapegoat for WHY you are doing these things.
The best way to do this is to engineer a crisis for your own people that would be just bad enough that no other monarch would believe that you did it to yourself: Turn your own people against you. That's right, you pay people to go out into your streets and sow discontent... but the kind of discontent that you want. Protests and maybe some riots start over all sorts of things like people wanting more education, new farm equipment, better infrastructure, etc. So, you spend everything you have and borrow a lot of money to meet the demands that you have engineered yourself to be good for the long term good of your nation. During this time you also bolster the heck out of your military which every other kingdom can clearly see you need right now.
Not only does this make you gold poor, but it also cements the eventually loyalty of your people. While other monarchs would send in their armies, to quell these sentiments, you helped your own people first and foremost, you proved that you are willing to hear and accommodate their concerns, and you've earned yourself the title "Queen Aerith the Kind" or something like that so that when the crisis is over, you will have the absolute loyalty of your own citizens.
Then, after about a year or two of crisis or whenever your mage slips up and talks, you reveal that you've just discovered gold alchemy, and you pay back all of your debts very quickly. Within weeks you flood foreign markets with tons of gold. Your own economy and military have become massively inflated to deal with your "crisis", and your competitors are now buried in undervalued gold.
Step 3, the truth now sounds like a lie
To the rest of the world, you were just in the right place at the right time, and if the mage does eventually slip up and say he invented the gold thing long before it was made known, well, everyone already believes he is a loud mouthed braggart. Only the most radical conspiracy theorists would believe that a queen intentionally turned her own people against herself based on the word of such a man.