Well, the concept of zero began to appears in India in Classical times, but I think it took a while before it was really formalized and widesrpead.
Even for China, that was closer, it took a similar time to really impose itself.
So, I'd think transmission trough Indias is a bit too unplausible, even if a less strong Persia could have meant stronger commercial relationship between Indias and Rome, and possibly culturally-wise.
Now, what could be more doable would be having a zero, not as a number, but as a vacant position as it nearly was during the imperial era on Greek numeral script (latin numeral script ignored it, and was frankly not really fit for mathematicals subtilities).
I suppose we could see such use blossoming by the Late Empire (especially trough administrative and fiscal needs) and *maybe* evolving later (principally trough indian influence) to a true zero.