If Einstein had carried a quartz watch instead of a light beam clock on his thought experiment spaceship, time dilation would still occur. Time dilation is a consequence of the theory of relativity and is a phenomenon that arises due to relative motion or differences in gravitational fields.
In the special theory of relativity, time dilation occurs when two observers are in relative motion to each other. If one observer is moving relative to the other at a significant fraction of the speed of light, they will perceive time to be passing more slowly for the other observer. This effect is known as time dilation and has been experimentally verified.
In Einstein's famous thought experiment, the light beam clock was used as a hypothetical device to illustrate time dilation. The concept was that as the spaceship moves near the speed of light, a light beam bouncing between two mirrors would take a longer path compared to an observer at rest. This longer path results in a slower ticking rate for the light beam clock relative to a stationary observer's clock.
If Einstein had carried a quartz watch instead of a light beam clock, the quartz watch would also experience time dilation. However, unlike the light beam clock, which is directly linked to the propagation of light, the quartz watch's operation relies on a stable oscillation of quartz crystals. The oscillation rate of a quartz watch would indeed be affected by time dilation, causing it to appear to tick more slowly when observed from a stationary frame of reference.
In summary, regardless of the type of clock used, whether a light beam clock or a quartz watch, time dilation would still occur in Einstein's thought experiment spaceship due to the relative motion involved.