I'm not sure if that's exactly what you mean by "being motivic", but this long exact sequence comes from a triangle in Voevodsky's category of (integral) motives $DM(\mathbb Q)$. 

More generally if $X$ is a $\mathbb Q$-scheme of finite type, $Z\subset X$ a closed subscheme, $\pi : Y \to X$ the blowup at $Z$, and $E=\pi^{-1}(Z)$, then there is a triangle
$$
M(E) \to M(Y)\oplus M(Z) \to M(X) \to M(E)[1]
$$
in $DM(\mathbb Q)$. This is proved in Mazza–Voevodsky–Weibel's [Lecture notes on motivic cohomology](http://www.claymath.org/library/monographs/cmim02.pdf), see equation (14.5.3). It can also be viewed a formal consequence of the six-functor formalism for $DM(-)$, as described by Cisinski and Déglise in their paper [Integral mixed motives in equal characteristic](https://www.math.uni-bielefeld.de/documenta/vol-merkurjev/cisinski_deglise.pdf).