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Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra of continuous functions on ordinary torus and by inserting a deformation parameter \theta into the product obtain a deformed non-commutaive C^*-algebra of functions on the quantum torus.

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Is there any natural way for deformation quantization of closed Riemann surfaces with higher genus in the above sense?

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One should definitely take a look at the work of Bordemann, Meinrenken, and Schlichenmaier: they provide a Berezin-Toeplitz inspired deformation quantization for all compact quantizable (i.e. the Kähler form is integral) Kähler manifolds. The asymptotics of this was discussed before by Cahen, Gutt, and Rawnsley in their 4 papers of quantization on Kähler manifolds. Later, Schlichenmaier and Karabegov determined the characteristic classes of the resulting formal star products explicitly, relating the BT-approach also to other versions.

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See the paper Quantization of Multiply Connected Manifolds, by Eli Hawkins. arXiv link.

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