Is every pair of conjugate variables associated with a Fourier transform? For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0,  i\neq j$$
I know that the position space representation of the wave function is the Fourier transform of the momentum space representation. From what I know, this also applies for other variables, such as energy and time, but I never read anything like that about other pairs of variables.
On the other hand, the commutator of other variables, such as x-angular momentum and y-angular momentum is different from 0. Does this mean that:


*

*$J_x$ and $J_y$ are a pair of conjugate variables?

*There exists a Fourier transform between the two?

*For any two variables in quantum mechanics that have a commutator different from 0, 1. and 2. are true?

*In general (not necessarily in quantum mechanics), to every pair of conjugate variables there is an associated Fourier transform? 

 A: The meaning of conjugate variables is usually that they are related by the Fourier transform (other transforms are possible, but I am not sure they are interesting). In quantum mechanics, this is simply a change of basis in Hilbert space. The relation between energy and time is similar, but time is a parameter in quantum mechanics, not an observable (this is even true in relativistic quantum mechanics). The meaning of the energy-time uncertainty relationship is consequently quite different from that of position-momentum. 
A: It depends very much what you mean by conjugate pairs, especially in quantum mechanics.
In classical mechanics the most common definition is based on the Poisson bracket:
\begin{align}
\{P_i,Q_j\}=\delta_{ij} \tag{1}
\end{align}
and thus in this sense canonical transformations (which can be non-linear transformations in the original coordinates) take you from a set of conjugate variables to another.
Alternatively, one can use Darboux’ theorem and the definition of the (closed) canonical symplectic form
\begin{align}
\omega = \sum_i dp_i\wedge dq_i
\end{align}
to define the canonical pair $\{p_i,q_i\}$.
Since (1) is at the root of Dirac quantization, where the quantum commutator is equal to $i\hbar \times$(classical Poisson bracket),  it is sensible to think of two operators $\hat P,\hat Q$ as conjugate if their commutator is $i\hbar$.   
Note that canonical transformations involving non-linear functions of operators is a mine field because of non-commutativity issues.
The plot immediately thickens with energy and time, since time is not a quantum mechanical operator. (Deriving energy-time uncertainly relation can be tricky and requires care.)  Thus if two classical quantities are Fourier pairs, there is no guarantee they will be conjugate “quantumly”.
Moreover, there are examples - for instance $\phi$ and $L_z=-i\hbar\frac{d}{d\phi}$ - which “look” conjugate in that they satisfy the correct commutation relations but in fact have issues of operator domains.  $\phi$ and $L_z$ are Fourier pairs in the sense that a rotation about $\hat z$ is of the form $e^{-i\phi L_z}$.  (This is an Abelian group so some technical hurdles linked to harmonic analysis are avoided.)
The notion of conjugate varibles in QM is also muddled by the related notion of complementary variables.  One may naively understand this as follows.  Given the uncertainty relation between $\hat x$ and $\hat p$ one may (loosely speaking) state that, if everything is known about one of the observable, nothing is known about the other.  An alternative statement would be that, if you are in an eigenstate of one operator, then the probability of outcome for the other is constant.  In this sense $\sigma_x$ and $\sigma_y$ (and indeed $\sigma_z$) are complementary as the outcomes of measuring $\sigma_y$ in any eigenstate of $\sigma_x$ (for instance) are equiprobable: if you know everything about the measurements of $\sigma_x$ (they have no fluctuations on eigenstates of $\sigma_x$), then you know nothing of the outcomes of measurements of either $\sigma_y$ or $\sigma_z$ as all the outcomes are equally probable.  This is the starting point for the notion of mutually unbiased bases.
