There are two types of frequency drift, long term and short term. A long term average frequency of an atomic clock is defined by the quantum properties of an atom and therefore is the same for all clocks of the same type. For example, two cesium clocks would not have a substantial long term drift relative to each other in the absence of relativistic effects. Best modern atomic clocks can be in sync within one second in the current lifetime of the universe.
A long term average frequency of a quartz clock is defined by the geometry of the quartz crystal that depends on the crystal physical dimensions and temperature. For this reason, no two quartz clocks have exactly the same frequency. Even if you used a thermally stabilized quartz oscillator in a computer, the NTP protocol would still be required, because the long term average frequency of your clock is unique and cannot match the frequency of the official atomic clock standard. The best quartz oscillators can be made of the same size within approximately one part in a million or one second in several days.
The second type of drift is a short term drift called jitter. The best known oscillators with the minimum possible jitter are thermally stabilized quartz crystals. Their jitter is much lower than the jitter of atomic clocks. There exist combination clocks where the long term stability of the atomic clock is further stabilized short term by a thermally stabilized quartz oscillator (or vice versa).