Sometimes when the weather is cold, my feet feel cold and uncomfortable while I am sitting at my desk. How could I warm up my feet while sitting at my desk?
Cold feet is typically a sign of decreased circulation, so whatever you do should aim at increasing the blood flow and probably generate some heat by muscle movement.
The most unobtrusive (but with limits) method is to wriggle and flex your toes. Do "grabbing" motions or stretch and lift the whole foot.
If you wear tight shoes or tend to sweat a bit at the feet, check whether switching to sandals or at least slipping out of your shoes for a bit is an option - if you sit a lot, nobody will notice your feet. Open or no footwear will let sweat evaporate (bonus: no stink) and not interfere with circulation. And fidgeting (see above) is easier. Without shoes, you can even place a tennis ball or other small object under your desk to grab and manipulate with your feet.
If moving the feet alone does not suffice, try to get your whole body in motion. Stand up (e.g. while on the phone), stretch, do some light exercises. If you are glued to your desk all day, your whole body will benefit, not just your feet. Consider walking to the kitchen to grab a cup of coffee, an extra bathroom break, walk to a collegue's desk instead of calling them. Still very unobtrusive. I have in the past even climbed a flight of stairs or walked briskly around the office - if it's large enough, everyone will assume you are on your way to or from somewhere. And try to schedule a walk outside during your lunch break. The oxygen will also clear your brain ;-)
Speaking of lunch: you could try to choose hot and spicy foods. They have a tendency to "heat up" the whole body. But the degree of effectiveness varies a lot between people.
Make sure your whole body is warm enough, so dress appropriately. No use trying to warm up your feet if the whole you is shivering.
And finally, if it really bothers you, consider getting a "warming cream": The type of over the counter medicine for sore muscles or rheumatic ailments containing capsicain or other substances that will locally increase the blood flow. Ask your pharmacist - you don't want a pain killer (often sold for the same use case). And if you do use it, make sure to wash your hands very well after applying, you don't want to get it inadvertently into your eyes, mouth or other mucous membranes.
I encountered this problem many years ago while working in a particular office. I tried multiple pairs of socks etc - but nothing worked. I assume there must have been some sort of cold breeze blowing at floor level because the rest of the office was perfectly fine.
Eventually, I bought myself a tube heater:
I plugged it into a spare power outlet under my desk and left it switched on most of the time. When my feet felt cold, I would remove my shoes and put my feet on the heater - the temperature never goes very high so there's no danger of burning, but be cautious the first time you do it.
Most importantly, the heater made a dramatic change to the temperature under the desk - I assume because it was a fairly "sealed" environment. It meant that, most of the time, I didn't need to put my feet on it at all.
They are cheap to buy and they use very little power. Depending upon your employer, you might be able to get them to buy one for you - but I reasoned that there was a fair chance that, if I asked, I'd be told it wasn't allowed. So I just did it myself and bypassed the red tape.
Take a thick 'throw blanket', fold over the bottom 10"-14" and sew the sides together to form a pocket for your feet. There's one under my desk right now, and it works great.
I find it helpful to change my socks in the middle of the day. Also, I heat up a cherry pit heating pad in a microwave and put it on or under my feet (slipped out of the shoes, of course). Alternatively you could use a hot water bottle. (Dry the outside well before use.)
Experiment with your shoes -- you want something breathable to prevent excessive sweating. However, if your office is drafty, then you don't want the super porous mesh sneakers, either.
If none of that does the trick for you, there are shoe warmer insoles you can buy. You plug them in to warm them up every few hours. If you buy these, and are unsure of the width, go for the narrowest option.
The best method I've found is also the cheapest and easiest. Make yourself - or buy - a wheat bag. A wheat bag is a simple cotton bag filled with whole grain wheat kernels. You can use any one of several different grains, but I'm most familiar with wheat. Take your wheat bag and put it in the microwave for a couple of minutes to warm up the wheat kernels inside. The bag comes out of the microwave deliciously warm and will stay that way for 20-30 minutes. I've put them on top of my feet at the office or down into my bed a few minutes before getting into a cold winter bed. The wheat kernels get warm from the microwave heating up the small amount of moisture inside.
Like I said, you can use several different types of filling if you want. Corn, rice, barley, oats, beans can all be used as long as they are raw, whole kernels. I've also seen suggestions to put essential oils or aromatic leaves in the bag as well but I prefer unscented myself.
There are several online tutorials on how to make wheat bags so just do a quick Google search and you'll find lots of resources!