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The sear modulus has no time dependence while the shear relaxation modulus has a time dependence. Is there a way to obtain G from G(t)?

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According to Rubinstein (Polymer Physics ISBN 9780198520597 section 7.6.1) the shear modulus is $G=\lim_{t\to \infty} G(t)$ which takes some non-zero value for solids.

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