What is correct use of 's

Both could be correct depending on the scenario.

Scenario 1: Your birthday. You got a car from your father, and a laptop from your mother. You could say- this is my father's gift, and that's my mother's gift. Though, a better way to say this would be- this is the gift from my father, and that's from my mother.

Scenario 2: Your father's birthday. You bought a cake and a gift. You could say- this is my father's cake, and that's my father's gift. Though, a better way to say this would be- this is the cake for my father, and that's the gift for him.

My father gifted me a car.

Having that context in mind, is it correct to write the statement below?

"Father's Gift"

Yes, it is correct because you have given sufficient context to make it clear.

Example

I was overjoyed when Father/my father gave/gifted me a car for my birthday. Father's gift enabled me to go to and from the sports ground whenever I needed.

Note that it is more usual to say "Father/Father's gift" amongst members of the family. To a stranger you would likely say "my father/my father's gift". This is because, in the former, Father is being used as a name. In the latter my father is used as a description.