A system of homogeneous linear equations always has the solution $ x=(0,\dots, 0) $. Suppose we have a system of $ n $ homogeneous linear equations in $ k $ variables. If $ k > n $ then there will always exist some non-zero solution. But what if $ k \leq n $? Assuming the equation are linearly independent then $ k \leq n $ implies that there are no nonzero solutions (rank-nullity theorem).
A system of quadratic forms $ q_1(x), \dots, q_n(x) $ always has the solution $ x=(0,\dots, 0) $. Suppose we have a system of $ n $ quadratic forms $ q_1(x)=0, \dots, q_n(x)=0 $ in $ k $ variables. If $ k>n $ then there will always exist some non-zero complex solution (indeed at least $ 2^n $ many (projective) solutions by Bezout's theorem). But what if $ k \leq n $? Can we conclude that no non-zero solution exists to the system?
At first it seems the answer is no. For example consider the system with $ n=k=2 $ consisting of the equations $ xy=0,x^2=0 $. However these equations aren't really independent since they have a degree one common divisor $ x $. But what if all the quadratic forms are independent? Then can we conclude that no solutions exist?
Question: Given a system of $ n $ independent quadratic forms in $ k $ variable with $ k \leq n $ is it true that no non-zero solution to the system exists?
I am not sure about what you are considering by "an independent system of quadratic forms". Considering that quadratic forms can be taken in any field, there is an entire chapter in Pfister's Book "Quadratic Forms With Applications to Algebraic Geometry and Topology" devoted to System of Quadratic Forms (Chapter 9).
There, Lemma 1.3 tells that if $char(K)\ne2$ and $(q_1,...,q_r)$ is an anisotropic system over $K$ then $q$ is regular. Which provides the answer "no" for your question if we consider "solutions" in the field K.
For considerations about the behavior of a system $(q_1,...,q_r)$ over $K$ under a field extension $K|L$ consult Pfister's book.
For a more detailed answer you need to specify the base field for the quadratic forms (or if you are considering only quadratic forms over $\mathbb R$) and what kind of "solutions" do you need.