Watson has been ruled out of the first Test in Brisbane after suffering the problem while bowling for New South Wales against Queensland in their Sheffield Shield clash on Saturday.
The all-rounder completed just one over and his absence is a major setback for the hosts as they prepare to lock horns with the No 1 Test side in the world.
Quiney staked his claim for a place by making 85 in Australia A's first innings of their three-day game against South Africa and is likely to bat at No 3.
Australia coach Mickey Arthur is hopeful that Watson will recover in time for the second Test at the Adelaide Oval on 22 November.
"We just think (Watson) will not be fit enough on Friday," Arthur said.
"We didn't want this to play out the whole week, we wanted to make a decision pretty early and get some real clarity in the team.
"That means Rob Quiney makes his debut this weekend against South Africa."
Arthur is confident veteran batsman Ricky Ponting will be fit to face the Proteas after hamstring trouble.
"We're very up-beat, it's a Test match week, a very special week. We're really excited," Arthur said.
"We're relaxed, we're calm, the team is in a really good space, we're really hungry for some success. We see this as a massive opportunity for us.
"In an ideal world, you want Shane Watson batting and bowling for you. Fact is, we're not going to have him. Fact is we won 4-0 against India last year without Shane."