Why do you want to push a car with the transmission in first gear?
If you push it in neutral, it will push a lot easier.
If the point is to turn the engine to make it start, because the starter is not working, then you could push it and let the clutch pedal up with the ignition on, and the engine may start.
Why you would want to push a car with the transmission in gear, when the point is not to start the engine, is strange.
If you are just asking if turning the engine by pushing a car with the manual transmission in gear will damage the transmission, then the likely answer is no (within sensibility).
But if you pushed the car 60 mph in first gear, then there would likely be damage to the engine.
I'm just not sure the context of your question.
If the engine can move the car through the transmission, then the car can move the engine through the transmission.
-Engine overhaul mechanic and general automotive mechanic since 1972