Put T = 1 hour. Then in the rest frame of an observer on the ground, the spacecraft will travel distance of v T, during a time of T, this means that the space-time interval between two points on the trajectory a time T apart is given (in c = 1 units)by
s^2 = T^2 - v^2 T^2 = (1-v^2)T^2
The space-time interval is an invariant, it will yield the same value if evaluated in the rest frame of the spacecraft. In that frame the spacecraft is at rest, while the time between the events is what we want to evaluate, let's call this T'. We then have that:
s^2 = T'^2.
We thus see that:
T' = sqrt(1-v^2) T
For small v we can expand this as:
sqrt(1-v^2) = 1 - 1/2 v^2
The ship's clock thus differs from one hour by:
-T/2 v^2
Restoring c can be done by recognizing that in c = 1 units you are free to multiply and divide by whatever powers of c you like because it is equal to 1 anyway, but of course, one particular choice will coincide with an equation that in SI units would also be dimensionally correct. So, we need to divide by c^2:
-T/2 v^2/c^2 = -2.4*10^(-2) s