Well, actually, the "motivic Haar measure" LSpice refers to is an analogue of Haar measure
that lives on ${\operatorname{GL}}_n({\mathbb C}((t)))$, not on ${\operatorname{GL}}_n({\mathbb Q}_p)$, and takes values in the Grothendieck ring of varieties, so I think it's not quite relevant here. What is closer to this discussion though is the fact that the usual Haar mesaure on ${\operatorname{GL}}_n({\mathbb Q}_p)$ if it's reasonably normalized (e.g. so that the volume of ${\operatorname{GL}}_n({\mathbb Z}_p)$

is $1$), actually takes values in $\mathbb Q$ on all reasonable sets that you ever want to consider. More precisely, one can define a sigma-algebra of the so-called "definable sets", and the volumes of definable sets just are in $\mathbb Q$. In this sense they are in
$\mathbb Q_p$ already, so the trick is that for these sets you do not need any completion of $\mathbb Q$ in order to define their volumes, and so you do not need to worry about using the $p$-adic metric... Most sets one works with turn out to be automatically definable, so this fact may be handy in some other situation.

Added some hours later: It was my first post on mathoverflow, and I am still not allowed to add comments to others' posts :) -- so this should be a comment to the comment by LSpice that appears in Emerton's post.
Talking about measure on $\operatorname{GL}_n(\overline{{\mathbb Q}_p})$, there is a paper by E.Hrushovski and D. Kazhdan http://arxiv.org/abs/math/0510133 that talks about integration in algebraically closed valued fields (using logic). As a first approximation, as far as I understand, the values of this measure are something like equivalence classes of definable sets over the residue field (I am certainly being imprecise here). There are several papers by Yimu Yin (the ones to start with are http://arxiv.org/abs/0809.0473, and http://arxiv.org/abs/1006.2467) aimed at clarifying this fundamental work of Hrushovski and Kazhdan in a slightly simplified setting. Unfortunately, I do not know of any non-technical introductory paper about this. There is a short note by Moshe Kamenski http://www.nd.edu/~mkamensk/lectures/motivic.pdf -- maybe this is the best place to start. Also, I hope someone corrects me here if I made any errors in this description.