Black hole thermodynamics establishes a deep and satisfying link to gravity, thermodynamics, and quantum theory. And, the thermodynamic property of black hole is essentially a quantum feature of gravity. In this paper, in order to study the influence of the quantum gravity effect on the quantum properties of black hole, we study the thermodynamics and its quantum correction to a non-commutative black hole. First of all, the temperature of the non-commutative Schwarichild black hole is calculated by using three different methods: surface gravity, tunneling effects and the first law of black hole thermodynamics. It is found that the same hole temperature is obtained by means of the surface gravity and tunneling effects. However, by using the first law of black hole thermodynamics, different results are derived from the first two methods. Therefore, we incline to the result obtained by surface gravity and tunneling effects, and the temperature obtained by the thermodynamic law needs modifying. That is, for the non-commutative black hole, there is a contradiction to the first law of thermodynamics. To calculate the temperature and other thermodynamic quantities for the non-commutative Schwarichild black hole, we use the corrected first law of black hole thermodynamics proposed in the literature. It is found that the black hole temperature derived by the corrected first law is the same as the temperature obtained by the surface gravity and the tunneling model, and the black hole entropy still follows Beckenstein-Hawking area law. Also, the heat capacity of the black hole is obtained and analyzed. It is seen that when the horizon radius and non-commutative parameter satisfy the particular conditions, the heat capacity is positive and the non-commutative black holes are thermodynamically stable. This is a different result from that of the usual Schwarichild black hole. Further, by studying the influence of generalized uncertainty principle on non-commutative black hole thermodynamics, the quantum corrections from generalized uncertainty principle for temperature, entropy and heat capacity of the non-commutative Schwarzschild black hole are given. It is found that with considering this quantum gravity effect, the obtained black hole entropy contains the item of are alogarithm. If the effect of the generalized uncertainty principle is neglected, the corrected black hole entropy can return to that in the usual case of Beckenstein-Hawing area law. Similarly, the corrected black hole temperature and heat capacity can also return to their counterparts in the case of usual Schwarzschild black hole when this quantum gravity effect is ignored.