In the context of e.g. a pseudo-orthogonal Lie group

$$\tag{1} O(p,q)~:=~ \{\Lambda\in {\rm Mat}_{n\times n}(\mathbb{R}) ~|~\Lambda^T\eta\Lambda= \eta \} $$

of pseudo-orthogonal matrices $\Lambda$ for the metric

$$\tag{2} \eta_{\mu\nu}~=~{\rm diag} (\underbrace{1,\ldots,1}_{p~\text{times}},\underbrace{-1,\ldots -1}_{q~\text{times}}), \qquad n~=~p+q,$$

a "vector of $O(p,q)$" is an element of the $n$-dimensional vector representation (aka. the defining representation or fundamental representation) of $O(p,q)$.

See also this related Phys.SE post.

In the context of e.g. a pseudo-orthogonal Lie group

$$\tag{1} O(p,q)~:=~ \{\Lambda\in {\rm Mat}_{n\times n}(\mathbb{R}) ~|~\Lambda^T\eta\Lambda= \eta \} $$

of pseudo-orthogonal matrices $\Lambda$ for the metric

$$\tag{2} \eta_{\mu\nu}~=~{\rm diag} (\underbrace{1,\ldots,1}_{p~\text{times}},\underbrace{-1,\ldots -1}_{q~\text{times}}), \qquad n~=~p+q,$$

a "vector of $O(p,q)$" is an element of the $n$-dimensional vector representation (aka. the defining representation or fundamental representation) of $O(p,q)$.

See also this related Phys.SE post.