No, a clock at rest with respect to Earth will not exhibit time dilation. Time dilation occurs when there is a relative motion between the observer and the object being observed, or when there is a significant difference in gravitational potential between the observer and the object.
In the case of a clock at rest with respect to Earth, there is no relative motion between the clock and the observer on Earth's surface. Therefore, from the perspective of an observer on Earth, the clock would not experience any time dilation effects.
However, it's worth noting that if the clock were placed in a region with a significantly different gravitational potential (such as near a black hole or in orbit around a massive object), then there would be gravitational time dilation. Clocks in stronger gravitational fields would appear to run slower compared to clocks in weaker gravitational fields. But in the context of an everyday clock on Earth's surface, the effects of time dilation are negligible and not noticeable to the observer.