The basic question is rather short: for a lattice $\Lambda_\tau = \mathbb Z + \tau \mathbb Z$, one can define the Eisenstein series $G_k(\tau) = \sum_{\lambda \in \Lambda_\tau \setminus \{0\}} \frac{1}{\lambda^k}$.
Question: can someone give a list of say five $\tau \in \mathbb H :=$ the upper half plane $\{z\in \mathbb C: \Im(z)>0\}$ s.t. $G_4(\tau)$ and $G_6(\tau)$ are both integers?
Background: for $\tau \in \mathbb H$ (the upper half plane $\{z\in \mathbb C: \Im(z)>0\}$), one can cook up a lattice $\Lambda_\tau := \mathbb Z + \tau \mathbb Z$, from which one can cook up the (Riemann surface) complex torus $\mathbb T_\tau := \mathbb C/\Lambda_\tau$, on which one can define the Weierstrass function and its derivative ($g_2:= 60G_4, g_3:=140G_6$): $$\wp_\tau(z) := \frac 1{z^2} + \sum_{\lambda \in \Lambda_\tau \setminus \{0\}} \left(\frac{1}{(z-\lambda)^2} - \frac 1{\lambda^2}\right), \quad \wp_\tau'(z) = 4 \wp_\tau(z)^3 - g_2(\tau) \wp_\tau(z) -g_3(\tau),$$
using which we can define the biholomorphic group isomorphism $\big[z \mapsto [1:\wp(z):\frac 12\wp'(z)] \big]$ from $\mathbb T_\tau$ to its image in $\mathbb C \mathbb P^2$ (the complex projective plane).
My question boils down to finding some explicit examples of lattices (parameterized by $\tau$) so that the Weierstrass elliptic curve for that lattice has integer coefficients.
For "natural choices" of $\tau \in \mathbb H$, one can do some crazy integral/sum manipulations to get the values of $g_2$ and $g_3$, but they don't seem to be integers in those cases: Invariant $g_2$ in Weierstrass's elliptic function, What are values of Eisenstein Series G2 on some specific points?, https://mathoverflow.net/questions/340154/determination-of-special-values-of-eisenstein-series. The Wikipedia page also has a picture of the Weierstrass $\wp_\tau$ function with $g_2 = 1+i$ and $g_3=2-3i$, though unfortunately it doesn't explain the process behind making this picture.
If you expand your horizons a little bit about what integer lattices we can look at then this is pretty straightforward. We might find the $G_k$ originally by studying the Taylor series of $\wp$ (this is done in most introductory books on the topic), so recall that in this context $G_k (\tau )$ is defined by the series $\displaystyle{ \sum_{\lambda \in \Lambda'} \lambda^{-k} } $. Some observation and then computation will show us that the $G_k$ value for the lattice $\Lambda$ spanned by $<1,\tau> $ differs from the $G_k$ value for the lattice $c \Lambda $ spanned by $ <c, c\tau> $ by a constant multiple:
$\displaystyle{ \sum_{\lambda \in \Lambda'} (c \cdot \lambda)^{-k} = c^{-k} \sum_{\lambda \in \Lambda'} \lambda^{-k}} $.
Note that here $c$ can be any complex number.
As you cite it's known that for the lattice $<1, i>$ that $\displaystyle{ g_2(i) = \frac{\Gamma(1/4)^8}{16 \pi^4}}$. Can you see how the above observation about the $G_k$ value for a scalar multiple of a lattice allows us to get to what you want from this result? What $c$ should I select to get from the previously found $g_2(i)$ to a lattice where the $g_2$ value is 3?