Your question is asking whether two Brownian motions can both first hit zero simultaneously. In fact we can say something stronger; for $N$ independent Brownian motions, the set of times where each Brownian motion hits 0 are pairwise disjoint.
When $N=2$, this is equivalent to asking whether a standard two-dimensional Brownian motion starting at $(x_1,x_2)$ ever hits $(0,0)$; it is known that, almost-surely, it will not.
For $N>2$, by the union bound, the probability is less than the sum over $i \neq j$ of the probability that a BM starting at $(x_i,x_j)$ ever hits $(0,0)$, so this probability is also 0.