  The $q$-binomial coefficients are q-analogues of the binomial coefficients,
counting the number of $k$-dimensional subspaces in the $n$-dimensional vector
space $\mathbb{F}^n_q$ over $\mathbb{F}_{q}$. In this paper, we define a
Euclidean analogue of $q$-binomial coefficients as the number of
$k$-dimensional subspaces which have an orthonormal basis in the quadratic
space $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ using a poset
structure on these subspaces. We prove its various combinatorial properties
comparing with those of $q$-binomial coefficients. In addition, we formulate
the number of subspaces of other quadratic types and study some related
properties.
