Can I calculate the t-value only using mean and standard deviation? Can I calculate the t-value of some population, if I only know the mean and standard deviation of the data and not the actual data itself. If not, what are the values I should know to calculate the t-value of the data?

No, you also need the sample size, $n$.
$$
t=\dfrac{
\bar x -\mu_0
}{
s/\sqrt n
}
$$
(It is typical, but not necessary, to assume $\mu_0=0$.)
