The Perl compiler produces the same opcodes for both types of strings.

>perl -MO=Concise -e "$a = 'abc'";
6  <@> leave[1 ref] vKP/REFC ->(end)
1     <0> enter ->2
2     <;> nextstate(main 1 -e:1) v ->3
5     <2> sassign vKS/2 ->6
3        <$> const[PV "abc"] s ->4
-        <1> ex-rv2sv sKRM*/1 ->5
4           <#> gvsv[*a] s ->5
-e syntax OK

>perl -MO=Concise -e "$a = qq{abc}";
6  <@> leave[1 ref] vKP/REFC ->(end)
1     <0> enter ->2
2     <;> nextstate(main 1 -e:1) v ->3
5     <2> sassign vKS/2 ->6
3        <$> const[PV "abc"] s ->4
-        <1> ex-rv2sv sKRM*/1 ->5
4           <#> gvsv[*a] s ->5
-e syntax OK
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The only difference would be in compile time, and the difference should be infinitesimal.