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The residue field $K=\Bbb F_5(c_0)$ of the declared ring $R=\Bbb Q_5(c)$ is the field with $5^3$ elements, and all elements in $K^\times$ have order $5^3-1=124$. So we can ask for the Teichmuller lifts of elements of $K$, for instance, for the element $1+c_0$ the lift ist as follows.

sage: R.<c> = Zq(125, prec=5)
sage: K = R.residue_field()
sage: R
5-adic Unramified Extension Ring in c defined by x^3 + 3*x + 3
sage: K
Finite Field in c0 of size 5^3
sage: R.teichmuller( K(1+c) )
(c + 1) + (4*c^2 + 3*c + 2)*5 + (4*c^2 + 3*c + 1)*5^2 + 3*c*5^3 + (c^2 + 2*c + 4)*5^4 + O(5^5)
sage: _^124
1 + O(5^5)

Note: Check also if the following does a better job in the intentioned application:

R.teichmuller_system()