According to this link mercury can be a part of every disease or condition known to man, so who knows if it is more than a small possibility. At this point, the answer is really academic for you.
Of much greater importance will be how your hypothyroidism is going to be diagnosed and treated. We have many experienced and knowledgeable members here who will be happy to help with that if you want.
Let me just pique your interest by telling you that the very highest priority for you will be to find a good thyroid doctor. All else is in distant second place. It does not have to be an Endo. In fact they are frequently prone to the "Immaculate TSH belief" and only want to use "Reference Range Endocrinology", neither of which adequately works for many patients.
A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms without being constrained by resultant TSH levels. I think you can get some good insight about this from this letter written by a good thyroid doctor for patients that he sometimes consults with from a distance. The letter is sent to the PCP of the patient to help guide treatment.
Take special note of this statement and always keep it in mind.
"The ultimate criterion for dose adjustment must always be the clinical response. I have prescribed natural dessicated thyroid for your patient (Armour or Nature-Throid). These contain T4 and T3 (40mcg and 9mcg respectively per 60mg). They are more effective than T4 therapy for most patients. Since they provide more T3 than the thyroid gland produces, the well-replaced patient’s free T4 will be around the middle of its range or lower, and the FT3 will be high-“normal” or slightly high before the AM dose."
Hope that helps your understanding. If you want to post your thyroid test results and reference ranges, members will be glad to try and assess the adequacy of your testing and treatment (if you have started on meds). If you have, what is the med type and dosage?
By the way, are you a lefty or right handed player?
I, too, have amalgam fillings, containing silver and researched possibly replacing them. My research indicated that the removal could actually cause more harm than leaving them, because the silver could be dispersed throughout the body much easier, with drilling/grinding.
That said, as gimel stated, it's probably a mute point, because we are already hypo, so getting rid of the fillings is not going change that, even if there was a connection.