Here is a counterexample: Take $n=2$ variables $X$ and $Y$. Let $L_1,\dots,L_5$ be linear polynomials such that $$S:=\{ (x, y) \in {\mathbb R}^2 ~|~ L_i(x) \ge 0\}$$ is a pentagon inscribed in the unit circle. Furthermore set $P:=1-X^2-Y^2$. Assume we could write $P$ as the sum of a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$. Now $P$ vanishes at the vertices of the pentagon and each $L_i$ is nonnegative at these vertices. Therefore $Q$ vanishes also at the vertices. But being a nonnegative quadratic polynomial, $S$ is a sum of squares of linear polynomials which all have also to vanish at the vertices and therefore are identically zero. This shows that $S$ is the zero polynomial. Now notice that each of the $L_i$ and $L_iL_j$ is strictly positive on at least one of the vertices of the pentagon (at which $P$ vanishes, of course). Since $P$ is a nonnegative linear combination of the $L_i$ and $L_iL_j$, this shows that $P=0$.
If the set $S$ defined by the $L_i$ has non-empty interior, then the convex cone of quadratic polynomials which can be written as a globally nonnegative quadratic polynomial $Q$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$ is closed. In fact, this follows from a much more general result on truncated quadratic modules, see e.g. the book of Marshall cited below (Lemma 4.1.4). This implies that, in the above counterexample, even $P+\varepsilon$ for small $\varepsilon>0$ will fail though this polynomial is strictly positive on $S$.
However, there are a lot theorems going into the direction of what you want. You might want to have a look at the following books...
- Marshall: Positive polynomials and sums of squares
- Prestel: Positive polynomials
- Bochnak, Coste, Roy: Real algebraic geometry
- Basu, Pollack, Roy: Algorithms in real algebraic geometry
- Knebusch, Scheiderer: Einführung in die reelle Algebra
- Andradas, Bröcker, Ruiz: Constructible sets in real geometry
...and the following articles...
- http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf
- http://www.math.uni-konstanz.de/~schweigh/publications/purestates.pdf
- http://www.math.uni-konstanz.de/~schweigh/publications/sosdualsdp.pdf
Also the so-called "S-procedure" could be of interest for you.