Will slinky length be different on the moon?

Simple answer yes,

Think about taking two extreme cases :

How much does a slinky extend in a gravity-free space? None at all

How much would it extend if it was on perhaps Jupiter or even a black hole ?It should extend by a large amount.

Gravity does play a role.

For a typical slinky (wiki) and comparing the gravity of earth and moon, this will be the case as outlined by the other answers.

However, for a general spring (wiki) or weaker gravity fields, the spring may be in a state where it is fully contracted and the weight of its own mass is insufficient to stretch it at all. In that regime, gravity does not play a role.

If a slinky is hanging vertically in a gravitational field, the amount of stretch in any short section depends on the weight of the coil hanging below that section. Less gravity will produce less stretch.