Is it illegal for my place of employment to deny me transfer to a different department without reasoning?

Based on the facts you presented, it's impossible to even venture a guess as to whether you are being discriminated against because you are a woman. While you mention that you and another female employee were denied transfer to a particular department, I'm not sure that's much of a pattern and I don't know whether working in the "call center" qualifies you as a "technician." Nor do I know how many women employees there are in the company and what jobs they hold. The "conflict of interest" remark is not sufficiently cogent to warrant a response. I don't, of course, even know whether your boss was referring to your situation. And, frankly, I don't see the slightest hint of any issue that involves a conflict of interest. As for whether this could be discrimination because you "share a workplace with [your] significant other," I am not aware of such discrimination being unlawful. Perhaps some other creative attorney can suggest that it fits within discrimination based on sexual orientation (which is unlawful in some states), but I think that's a stretch. And as a NY attorney I don't know whether there would be any basis at all under your state's law. You might wish to inquire what the rules are, if any, for transfers. Good luck to you.