How to calculate the eigenvalues of a matrix over a finite field?
The characteristic polynomial of that matrix factors over $GF(5)$ as $$ \chi_A(x)=\det(xI_5-A)=\left(x^2+2 x+4\right) \left(x^3+2 x^2+2 x+2\right). $$ Therefore the matrix $A$ has two eigenvalues in $GF(5^2)$ and three eigenvalues in $GF(5^3)$. From the first factor $$x^2+2x+4=(x+1)^2-2$$ we see that $-1\pm\sqrt2$ are the eigenvalues in $GF(5^2)=GF(5)(\sqrt2)$.
If you want a single field to contain them all you need to go to $GF(5^6)$.