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CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it in Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

This also works for 22 bytes (printing the same quines):

"& *NQ!":)R"{'_'~}_~";

Explanation

In CJam:

"'!+OR'"    e# Push this string.
)           e# Pull off the last character.
5-          e# Subtract 5, turning ' into ".
"{'_'~}_~"  e# Push this string.
;           e# And discard it again.

So at the end of the program, the stack contains the string "'!+OR" and the character ", both of which are implicitly printed.

In Fission, program flow starts at the R with a right-going atom. '" just changes the atoms mass, ), 5 and - are ignored for various reasons. Then the atom enters print mode at the " and prints {'_'~}_~. ; destroys the atom and terminates the program.

CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it in Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

This also works for 22 bytes (printing the same quines):

"& *NQ!":)R"{'_'~}_~";

Explanation

In CJam:

"'!+OR'"    e# Push this string.
)           e# Pull off the last character.
5-          e# Subtract 5, turning ' into ".
"{'_'~}_~"  e# Push this string.
;           e# And discard it again.

So at the end of the program, the stack contains the string "'!+OR" and the character ", both of which are implicitly printed.

In Fission, program flow starts at the R with a right-going atom. '" just changes the atoms mass, ), 5 and - are ignored for various reasons. Then the atom enters print mode at the " and prints {'_'~}_~. ; destroys the atom and terminates the program.

CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

Explanation

In CJam:

"'!+OR'"    e# Push this string.
)           e# Pull off the last character.
5-          e# Subtract 5, turning ' into ".
"{'_'~}_~"  e# Push this string.
;           e# And discard it again.

So at the end of the program, the stack contains the string "'!+OR" and the character ", both of which are implicitly printed.

In Fission, program flow starts at the R with a right-going atom. '" just changes the atoms mass, ), 5 and - are ignored for various reasons. Then the atom enters print mode at the " and prints {'_'~}_~. ; destroys the atom and terminates the program.

CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it in Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

This also works for 22 bytes (printing the same quines):

"& *NQ!":)R"{'_'~}_~";

Explanation

In CJam:

"'!+OR'"    e# Push this string.
)           e# Pull off the last character.
5-          e# Subtract 5, turning ' into ".
"{'_'~}_~"  e# Push this string.
;           e# And discard it again.

So at the end of the program, the stack contains the string "'!+OR" and the character ", both of which are implicitly printed.

In Fission, program flow starts at the R with a right-going atom. '" just changes the atoms mass, ), 5 and - are ignored for various reasons. Then the atom enters print mode at the " and prints {'_'~}_~. ; destroys the atom and terminates the program.

CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

I'll add an explanation in a bit.

CJam / Fission, 22 bytes

"'!+OR'")5-"{'_'~}_~";

Try it in CJam. Try it Fission.

In CJam, this prints the standard Fission quine:

'!+OR"

Try the Fission quine.

In Fission, this prints a "-less variant of the standard CJam quine:

{'_'~}_~

Try the CJam quine.

Explanation

In CJam:

"'!+OR'"    e# Push this string.
)           e# Pull off the last character.
5-          e# Subtract 5, turning ' into ".
"{'_'~}_~"  e# Push this string.
;           e# And discard it again.

So at the end of the program, the stack contains the string "'!+OR" and the character ", both of which are implicitly printed.

In Fission, program flow starts at the R with a right-going atom. '" just changes the atoms mass, ), 5 and - are ignored for various reasons. Then the atom enters print mode at the " and prints {'_'~}_~. ; destroys the atom and terminates the program.