Can Hypothyroidism medication cause depression?

Uncommonly yes. I'm not sure WHICH medication you're referring to but if we just hazard a guess and say, "LEVOTHYROXINE" then, in rare instances this medication has been reported to cause mental depression in patients. Typically though, when used in therapeutic doses to treat hypothyroidism one usually experiences a bit of a lift in mood (if they've been truly HYPO-thyroid) for some time.
Yes. Both hypo and hyperthyroidism can be accompanied by depression. If you have been hypothyroid and are on replacement therapy you must have your levels checked to assure proper dosing. If your thyroid condition is well treated, your depression is not related to it.