Your question has two parts and the answers are Yes, and Maybe.
Yes, the lender will consider a gift of equity from a family member as down payment.
Maybe, the lender will not require a cash down payment. If you are doing an FHA loan you will not be required to provide an additional cash down payment, but keep in mind that someone will need still need to pay closing costs. If you are doing a conventional loan and the down payment isn't at least 20% of the appraised value or sale price, you will need to provide a down payment of as much as 5% from your own funds.