Can you Wear Hats Indoors in Winter?
There are a few things to think about with regard to this question:
Where do you work - is it appropriate in that environment?
In a more creative environment, it may be fine to wear a hat of some sort, whilst in a conservative or traditional environment, it would not.
What Kind of Hat?
There is one kind of hat that is the most likely to work inside in winter, and that's the beret, this is because in some ways it can look more like an accessory (like a scarf) rather than a hat. Beanies may also work, but they are harder to wear for many!
Hats with brims are harder to wear inside as the brim partially shields your face, which may obscure your eyes some and reduces your ability to communicate with others. An upturned brim, like on a fedora is a much better option than a downturned brim, like on a cloche or bucket hat.
What Colour Hat?
If you really want to wear a hat inside in winter, I'd choose one without a brim, and either make it a fashion statement (as a feature accessory in your outfit) or choose one in the colour of your hair, so that it blends with your hair and is less obvious and obtrusive.
How You Wear the Hat
Ideally, when wearing hats indoors, style them so they sit further back on your head rather than forward and covering your eyes. Keep your face free as you can of the hat so that others don't feel that the hat is obscuring your face and your ability to communicate freely.
There is no right answer, but these are my thoughts and you can try them out for yourself. Sometimes wearing a hat can feel unfamiliar but you fairly quickly get used to it, other times you may feel that wearing a hat is like a costume, and it's not authentic to you. Different styles of hats may elicit different internal responses and only you know the answer of what is right for you!
More Tips on Choosing Hats
How to Accessorize in Winter
How to Choose a Hat to Flatter
How Hats and Headwear Communicate and the Personality of Different Styles
How to Accessorize with Hats, Scarves and Gloves