Let $X$ be a strict $\infty$category (not $(\infty,1)$, I am talking about true $\infty$categories (Grothendieck modules (exact presheaves (finitelimit preserving functors $\Theta^{op}\to \mathrm{Set}$) over Joyal's category $\Theta$ (see Dimitri Ara's thesis)). Is there a notion of a Grothendieck fibration between two such $\infty$categories or a notion of a 1cell in inside such a strict $\infty$category being a Grothendieck fibration? If so, how is it formulated?
It depends on whether you want to mimic the 1categorical case strictly, or take into account the homotopy nature of the $\infty$categories. Clearly (or I would hope so!) there is an underlying 1category of an $\infty$category, and so a trivial way of arriving at a fibration. I guess this is not what you are after. My guess would be to imitate the definition along the lines of what I think Street gave in 'Fibrations in bicategories', i.e.
where $f/_{\cong} p$ is the 2pullback, and $f/p$ is the comma object (from the nLab). So we should just replace 2categorical concepts here by $\infty$categorical concepts. This presumes the definition of adjoint $\infty$functors :) As to your second question, if we know that $\infty$Cat is enriched over itself, we use the $\infty$categorical version of the first definition at fibration in a 2category (nLab again) EDIT: Since you are after Grothendieck fibrations, not Street fibrations, you would want the $\infty$versions of strict 2pullbacks and strict comma objects. This should be a significant simplification. I"m not sure, but you may have strict $\infty$pullbacks = strict $\infty$isocomma object, and if both the comma and isocommaobjects can be constructed using cotensors, then to me this would be the best approach. But it of course depends on what you want to do next (aside: yes, I saw your other post in that other place). 


I don't know of anyone having written down such a definition for the particular model of ∞categories that you appear to be interested in, but the general form of the notion of fibration for higher categories is known at an imprecise level. See for instance the nLab page on nfibrations. 

