Doing pushups, and the fact that it can be socially awkward to do in the office, are both good and valid points. If you are lucky, people won't be able to see you do them. But there are plenty of other things you can do in the same spirit just while sitting, such as 1) leg lifts (both straight and bent knee), 2) kegels, 3) butt clenches, 4) calf stretches (foot flat on ground and bend toes up, or go other way and keep toes on ground and bend heels up), 5) squeeze legs together, 6) or, similar to kegels, simply just flex any muscle (hold it, or pulse flexes).
I'm sure you can come up with others as well. Just be creative. This is hardly going to do anything for strength and flexibility, and it will only be a very weak cardio workout, but its discreet, can be done while working, and nobody will notice. Do it enough so that you can do various moves throughout the work day without paying much attention to it (and thus not affecting your work much), and you'll be feeling great without being "that sweaty, awkward push-up guy"