We know that the evaluation functor $\Gamma(u, -):Qcoh(X) \to \{\cal O}_X(u)-mod$ is a left exact functor preserving limits. We also know that the category of all quasi-coherent sheaves on scheme is locally finitely presented. So we may use "Adjoint Functor Theorem" and deduce that there is a left adjoint for the evaluation functor.

Is there any explicit description for this adjoint?

The answer is true if we replace $Qcoh(X)$ by the category of sheaves on $X$ (see section 2 of "Relative Homological Algebra in Categories of Representations of Infinite Quivers" by S. Estrada)

can notbe written as some category in section 2 of the paper you cite. Even if we fix the representation of rings O_X, we cannot write Qcoh(X) as a category of representations of modules on this. This is claimed in "Relative homological algebra in the category of quasi-coherent sheaves", but it is wrong because no compatibility data can be imposed on a quiver. Instead, one has to use categories (in this case the category of open affine subsets of X). This is rather unfortunate since many people cite this paper and take this claim for granted ... – Martin Brandenburg Mar 10 '12 at 9:41