Do these definitions of integrable quaternionic structure agree? I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.
One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic structure implies that there exists coordinates in each coordinate neighborbood whereby the three almost complex structures $J_1,J_2,J_3$ (that define the almost quaternionic structure) take the form
$$
   J_1= \left[\begin{array}{cccc}
    0 & -1 & 0 & 0 \\
    1 & 0 & 0 & 0 \\
    0 & 0 & 0 & -1\\
    0 & 0 & 1 & 0
    \end{array}\right]
$$
$$
   J_2= \left[\begin{array}{cccc}
    0 & 0 & -1 & 0 \\
    0 & 0 & 0 & 1 \\
    1 & 0 & 0 & 0\\
    0 & -1 & 0 & 0
    \end{array}\right]
$$
$$
   J_3= \left[\begin{array}{cccc}
    0 & 0 & 0 & -1 \\
    0 & 0 & -1 & 0 \\
    0 & 1 & 0 & 0\\
    1 & 0 & 0 & 0
    \end{array}\right],
$$
where $1$ is the identity matrix.
On the other hand, from this  article, I have found a definition where integrable quaternionic structure implies that the Nijenhuis tensor for each of the almost complex structures given above vanish. 
Do these definitions agree? 
 A: These two 'definitions' do not agree.  Also, you should be careful about your choice of sources.  Most differential geometers use the terminology 'almost quaternionic' to mean that the structure group of a $4n$-manifold $M$ has been reduced to a subgroup of $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)\subset\mathrm{GL}(4n,\mathbb{R})$.  So, for example, $\mathbb{HP}^n$ is a(n integrable) quaternionic manifold in this sense, even though it does not have any almost complex structures.  When the structure group has been reduced to  $\mathrm{GL}(n,\mathbb{H})\subset\mathrm{GL}(4n,\mathbb{R})$, i.e., when there are $3$ anti-commuting almost complex structures defined on $M^{4n}$, then one says that $M$ is almost hypercomplex.  From your question, I assume that this latter term is what you mean by almost quaternionic.
In any case, the condition that there exist local coordinates in which the three almost complex structures have constant coefficients is the condition that the hypercomplex structure be locally flat, which is much more restrictive than the second condition, which is that each of the $J_i$ be an integrable almost complex structure (i.e., that its Nijnhuis tensor vanish).  For example, even in the much more restrictive case of hyperHermitian structures (i.e., when the structure group is $\mathrm{Sp}(n)\subset\mathrm{GL}(n,\mathbb{H})\subset\mathrm{GL}(4n,\mathbb{R})$) flatness is much more restrictive than the condition that the three almost complex structures be integrable.  
Any hyperKähler manifold is integrable in the second sense, but it is not integrable in the first sense unless the metric is actually flat.
For another kind of example, see The Hypercomplex Structure of $SU(3)$.
