In Nekovar's introductory paper "Beilinson's Conjecture"
http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf
The conjecture is formulated for smooth projective varieties over $\mathbb{Q}$. However all the statements and proofs seem to equally work even for smooth projective varieties defined over an arbitrary number field $L$. Is this true? Are there any differences between the case over $\mathbb{Q}$ and number fields $L$?
Yes, you can formulate Beilinson's conjectures for smooth projective varieties over a number field. I would recommend the following survey paper by Dinakar Ramakrishnan: Regulators, algebraic cycles and values of $L$-functions. Other references, with an emphasis on the equivariant version of the conjecture, include: Fontaine--Perrin-Riou, Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions $L$, and the articles by Burns and Flach: Motivic $L$-functions and Galois module structures; Tamagawa numbers for motives with (non-commutative) coefficients I, II.
I should add that if $X$ is a smooth projective variety over a number field $K$ then its Weil restriction $Y=\operatorname{Res}_{K/\mathbf{Q}} X$ exists and is a smooth projective variety over $\mathbf{Q}$, and the Beilinson conjectures for $X$ and $Y$ should be equivalent. I don't know a reference where this is worked out in detail, but see Fontaine, Valeurs spéciales des fonctions $L$ des motifs, Remarque 6.11 for some indications.
In addition to François's answer, I'll address the second question.
Are there any differences between the case over $\mathbb Q$ and a number fields $L$?
The main difference - which can be dealt with but should not be forgotten nevertheless - is that there are now many embeddings of the field of definitions into $\mathbb C$, and that some but not necessarily all of them might be real embeddings.
Since Beilinson's conjecture makes crucial use of the Hodge structure on the singular cohomology $H^i((X\times_{L,\sigma}\mathbb C)(\mathbb C),\mathbb Q)$ of the variety $X$ (with $\sigma:L\hookrightarrow\mathbb C$ an embedding) and more precisely of the Hodge structure on $H^i((X\times_{L,\sigma}\mathbb C)(\mathbb C),\mathbb Q)\otimes_{\mathbb Q}\mathbb C$ (which is always an $\mathbb R$-Hodge structure, but not always an $\mathbb R$-Hodge structure over $\mathbb R$) one must be careful when generalizing from $\mathbb Q$ to $L$: should one fix an embedding? consider them all at once? does the conjecture depend on this choice? what if some embeddings are real and some complex? etc.
Incidentally, the same difficulty arises for the Bloch-Kato conjectures predicting the exact value of special values of $L$-function with respect to the $p$-adic étale realization (in that case, there might be many primes above $p$ in $L$ and the $D_{\operatorname{dR}}$-module appearing in the conjecture might depend on the choice of the prime).
