No - hydroxychloriquine is used to "treat" the symptoms of Lupus, RA and a couple other autoimmune conditions. The only thing that can treat hypothyroidism is replacement thyroid medications, such as T4 medications containing levothyroxine (brand names include Synthroid, Levoxyl, etc), T3 medications containing Liothyronine (brand name Cytomel) and desiccated hormones that contain, both T4 and T3 (these include Armour, NatureThroid, WP, etc).
Although there are plenty of articles online about "curing" Hashimoto's, there is no real cure. Diet changes make a big difference for some people, but not for others. There is "nothing" that will repair thyroid damage once it's done. For as long as you've had hypothyroidism/Hashimoto's, it's probably pretty safe to assume that your thyroid is completely dead and produces nothing on its own. You'll have to stay on replacement medications for the rest of your life.
Just to make it perfectly clear - there is nothing that can "cure" an autoimmune condition, nor can anything reverse damage that's already been done. Hydroxychloroquine is used only to treat symptoms, not to eliminate or cure anything. These diseases attack various systems through the body (Lupus can attack any/all organs in the body), whereas Hashimoto's only attacks the thyroid. Since the thyroid produces thyroid hormones and nearly every cell in the body needs thyroid hormones, replacing them via daily medication is the only treatment.