Can I build muscle just by doing push-ups and deadlifts?

ABSOLUTELY!!!

1. Your chest (pectoralis) will develop from consistent push-ups. I promise you will notice a difference in a month if you remain dedicated. Just 100 push-ups a day will have profound results. If you are doing five sets of 20 push-ups now, you will be doing three sets of 33(.33) by then. Slow, steady gains.
2. Push-ups also are a great way to engage the core (abdomen) muscles. If you want to work on your entire upper anterior physique, push-ups are going to be your go to exercise.

As evidenced by this picture, deadlifts work a plethora of muscle groups. By dedicating time and energy to deadlifting at least twice a week, you should develop both your posterior and anterior leg muscles, as well as your back muscles & traps.

Be sure not to over exert, you don’t need to max out every time you lift & quite frankly, you shouldn’t!

Try to lift about 70% of your max around 5 reps for 4–5 sets.

I think you will notice tremendous results in doing so!

Hope this helps you reach your fitness goals :)