I need to prove that every element of Q (rational numbers) has an inverse with respect to addition. From what understand, I need to use the fact that [0,1] is the additive identity of Q.

0 is the additive identity of Q. And you can have 1 and only 1 additive identity.

As for a proof, I would just say let a/b be integers such that b is nonzero. Then a/b + -a/b = 0, and thus, every rational number has an additive inverse.

Also, I need to prove that every element of Q except [0,1] has an inverse with respect to multiplication. Again, to do so, I'm thinking it has something to do with the fact that [1,1] is the miltiplicative identity of Q.

[1,1] is better written as just 1. The only rational number that doesn't have a multiplicative inverse is 0. Consider a/b where a and b are integers and a and b are nonzero. Then a/b * b/a = 1, and thus, all nonzero rationals have a multiplicative inverse.