Classical logic in concern with QM Mathematics In no way am I a physicist, so please excuse improperly used terms.

It is in my understanding that Quantum Physics does not obey Classical Logic, hence the existence of Quantum Logic.
My questions are: 
If mathematics obeys Classical Logic how are we able to represent QM mathematically?  
Is QM represented through a different type of mathematics that obeys Quantum Logic instead of Classical Logic?
 A: The most elementary formulation of Quantum Mechanics (the one usually formulated in Hilbert spaces) can be constructed starting form a lattice of all the elementary propositions which can be tested on a given quantum system obtaining, as the outcome, YES or NOT. 
This can be similarly done for classical mechanics and, in that case the elementary propositions are described by a class of sets in the space of phases of the physical system, an elementary proposition $P$ is true (YES) if the state of the system belongs to $P$ at the considered time. That class of sets/propositions has to be closed under the action of logic operators/set operations. For instance $P$ AND $Q$ corresponds to $P \cap Q$,  and so on...
A state of the system is, in fact,  a map associating each elementary proposition to a number in $\{0,1\}$, where $0$ means NO and $1$ means YES.
 If instead the whole set of the outcomes $[0,1]$ is allowable, the state is probabilistic, it is the probability that a elementary proposition is true
(this is the standard, for instance, dealing with statistical mechanics).
The fact that a state is a probability measure (a Dirac measure for sharp states) forces the class of subsets describing propositions to be a $\sigma$-algebra, which is a notion a bit more complicated than a Boolean algebra.
In quantum mechanics one could try to adopt a similar point of view, singling out the elementary YES-NO propositions from scratch when describing a quantum system. The point is that there are pairs of elementary propositions that cannot be joined by connectives: They are the famous incompatible propositions. Here is a typical example. $P=$ "the particle has momentum $p$"  and $Q=$ "the particle has position $q$". There is nothing like P AND Q in Nature. No physical experiment  can associate a value YES/NO to $P$ AND $Q$.
Physicists say that $P$ and $Q$ are incompatible. A Boolean structure cannot be used here!
However the situation is not so desperate as it may seem at first glance, because there are mathematical structures (non Boolean lattices) able to grasp and  describe the physics of elementary propositions of a quantum mechanical system. As a matter of fact these structures are isomorphic to the lattices of orthogonal projectors  of Hilbert spaces. The states can be defined similarly to the classical case, (generalized) probability measures, and they turn out to be associated with the normalized vectors of the mentioned Hilbert spaces (I refer here to the so called pure states only).
The point is that these mathematical structures, non Boolean lattices, embody operations that just resemble the classical connectives AND, OR, NOT, $\Rightarrow$ and so on. They are generalizations of the classical corresponding and reduce to them when dealing with sets of pairwise compatible propositions.  
Here a twice possibility arises. One can ignore this fact and  exploit it as a merely technical opportunity. Alternatively one can assume that these pseudo connectives are the connectives of the quantum world which, in that precise sense (connectives with different properties), satisfy a logic different from the classical one.  This was the idea of von Neumann and Birkhoff, who started the first investigation of that mathematical world. Nowadays, quantum logic is a (wide) research area more proper of logicians rather than physicists. I mean, for several reasons even of practical nature, von Neumann and Birkhoff were not able to convince physicists to abandon classical logic in favour of (some) quantum version.
In any cases, there are no problems in handling quantum logic, because it is nothing but a formal procedure to construct statements when initial statements are given. To this regard it is not different from any other mathematically formalized theory. We can handle quantum logic using intuitive logic exactly as we can handle some formalization of classical logic using intuitive logic.
A: The first lines of the Wikipedia entry Quantum logic give an impression. To cut it short, the essence of the example is this: You have a particle smeared out in some box of length $d$. If you split $d$ into two parts, $d_\text{Left}$ and $d_\text{Right}$, then "the particle is in the union of $d_\text{Left}$ and $d_\text{Right}$" is true by definition, while "the particle is either in $d_\text{Left}$ or in $d_\text{Right}$" is classically equivalent but not sensible in quantum physics.
So we see aspects of quantum mechanics are not directly expressible using the propositional rules of classical logic - the language isn't well suited to talk about it straight away. 
See Nonfirstorderizability for a completely different example of where the language you'd like to use doesn't quite make it: "Some critics admire only one another." 
So while there is the possibility to do quantum logic which accurately captures the QM system directly, you don't have to. 
Mathematics is usually thought of as taking place in classical logic, very often extended with axioms about sets (most people don't care though), just because it suffices. 
For example, as the set $\{c,b,a,b\}$ is by defintion the same as the set $\{a,b,c\}$, this isn't so well suited to talk about lists of things. But people then go on and define the pair $(x,y):=\{\{x\},\{x,y\}\}$ and define the triple $(x,y,z):=(x,(x,z))$ and now they got a notion of list within set theory (don't ever speak about the ugly actual thingy "$\{\{x\},\{x,\{\{y\},\{y,z\}\}\}\}$" down at the hardware level). 
For whatever you want to say about particles in boxes, you might want to (or have to) use the more loaded framework of expectation values for probability distributions and whatnot... This is a theory written down completely classically. In any case, you can write down quantum mechanics in the formal language of classical mechanics and then start to reason about it -formally or informally- using quantum logic. It's a tool, at whichever state you want to apply it. 
But it's also noteworthy that quantum logic isn't the only non-classical logic language people try to approach quantum mechanics with. By the way, if you say "if mathematics obeys Classical Logic" then you either express that you're a Platonist of some sort, or you purposely narrow the scope of mathematics. Nonclassical logics are becoming more popular, I'd say. Let me also link to my very favorite answer on the StackExchange network: 
Is First Order Logic the only fundamental logic?.
And see also this question.
