I know the straightforward answer to this question is no: electrons are leptons which by definition don't interact via the strong force, gluons are the mediators of the strong force and hence electrons don't interact with gluons.
That said, in this power point by D.Stump the parton distribution function is defined as (pg45, paraphrased):
$f_i(x,Q^2)dx$ is the mean number of the $i$th type of patron with longitudinal momentum fraction from $x$ to $x+dx$ appropriate to a scattering experiment with momentum transfer $Q$.
My concern is that $f_0(x,Q^2)$ is in general non-zero, with the subscript $0$ here referring to gluon. From my interpretation of the phrase appropriate to a "scattering experiment with momentum transfer $Q$" I would interpret this to mean that the scattering particle (an electron) interacts with these quarks. Is this interpretation correct - if so does this mean that electrons interact with electrons and if not how should this phrase be interpreted.