No. The reason is that
$$e(f,Q)=\int_a^b \bigg\{f^4\bigg[1+\|\frac{d}{dr}Q\|^2+f^2\dot f^2\bigg]\bigg\} \,dr,
$$
with $b\geq f\geq a$ (since $f$ is continuous and monotonous) gives 
$$
e(f,Q) \leq \left(1+\frac{b^4}{2}\right)\left(\|f\|^4_{W^{1,4}} + \|Q\|^2_{W^{1,2}}\right)
$$  
So your inequality would imply that there exists $C_1,C_2>0$ such that for all $f,Q\in\mathcal{E}$
$$
C_1\|Q\|^2_{W^{1,2}} + C_2\|f\|^4_{W^{1,4}} \geq  \|Q\|^4_{W^{1,4}}.
$$
Now, fix $f=a+b\frac{t-a}{b-a}$, and it becomes for all $Q\in\mathcal{E}$
$$
C_3(\|Q\|^2_{W^{1,2}} + 1) \geq  \|Q\|^4_{W^{1,4}},
$$
if you cook-up a sequence $Q_n$  with $\|Q_n\|_{W^{1,2}}=1$ and  $\|Q_n\|_{W^{1,4}}>n$  you have a contradiction.  

Note that $W^{1,4}$ is not natural, since you can also obtain a bound from above of the form
$$
e(f,Q) \leq C(a,b)\left(\|f\|^2_{W^{1,2}} + \|Q\|^2_{W^{1,2}}\right).
$$