Does a Roth conversion count towards the $4,300 "income" limit for someone to qualify as "other dependents"?

Yes, this is income. It is moving the original contribution (pretax) to another IRA that is after-tax.

[Edited 4/13/2022|5:21 AM PST]

You mean money that was converted from a Trad IRA to a Roth?  Yes, it is taxable and counts towards the $4300.,