Thank you for the "nitpicks"! I just edited the question to make conventions explicit.

In the algebraic context, the answer is yes, even without properness. For projective varieties, one could use the theory of the Hilbert scheme. But there is an easier argument that applies to all schemes of finite type over $\mathbf{C}$: enumerate finitely generated field extensions of $\mathbf{Q}$ up to isomorphism, enumerate all schemes of finite type over these, and spread out each one to a family over an integral $\mathbf{Q}$-variety.

More generally, one could ask whether there exist countably many proper morphisms of complex analytic spaces such that every compact complex analytic space appears as a fiber in at least one of the families?

For some recent papers related to the work of Aubry and successors, see Clark, Euclidean quadratic forms and ADC forms: I. Acta Arith. 154 (2012), no. 2, 137–159; Clark & Jagy, Euclidean quadratic forms and ADC forms II: integral forms. Acta Arith. 164 (2014), no. 3, 265–308; and Dacar, Euclidean quadratic forms are ADC forms: a short proof, arxiv.org/abs/1310.2093